amp (or receiver) recommendation for Polk Audio RTi A7 floorstanding speakers

See here for their M30.1, the smallest that I think you might use comfortably in the smallest of your two potential music rooms: http://www.theabsolutesound.com/articles/harbeth-monitor-301-loudspeaker/
Or here for their M40.2, their biggest and most expensive: http://www.theabsolutesound.com/articles/harbeth-monitor-402-loudspeaker/

A 2x100 watt Yamaha AS 801 amplifier, with both optical and usb inputs, will be a more than good enough amplifier for these speakers. See here for a test of the AS 700, its predecessor without digital inputs: http://www.avhub.com.au/product-reviews/hi-fi/yamaha-a-s700-integrated-amplifier-393552 If you want to save some money, you can get an AS 701 (the same but without usb), but you will need a converter to convert one optical signal to coaxial digital. I would get the AS 801 for the extra digital input, so you can also add a Chromecast Audio streamer’s digital output if you ever decide to go that route (as I am sure you will one day do).
I am not sure I would go the two main speaker route. The resulting low joint impedance does put a bit of strain on the amplifier. Why not sell the A5’s?
A subwoofer is always a good idea, but needs some attention. At the very least I would use an Antimode 8033 room eq for much cleaner bass. I am very pleased with mine. See: http://www.theabsolutesound.com/articles/dspeaker-anti-mode-8033-dsp-subwoofer-equalizer-tas-204/
Low frequency response gets even better (smoother and cleaner) with a second sub: http://www.acousticfrontiers.com/20101029using-multiple-subwoofers-to-improve-bass-the-welti-devanti...
If you want to get an amplifier for a more ambitious system (particularly in that largest of the two rooms) and that still does not break the bank, think of a Pioneer U-05 pre-amplifier/DAC combined with a pro audio power amplifier like the recently discontinued 2x250 watt Yamaha P2500S or 2x350 watt P3500S pro audio power amplifiers. The Pioneer and the Yamaha both have professional style balanced connections for a lower noise floor. See here for a very favourable and rigorously scientific test of the latter model: http://www.homecinema-fr.com/forum/amplificateurs-de-puissance-haute-fidelite/mesures-ampli-yamaha-p...
This is what I would do, as it will have more digital inputs, and the power to drive any high quality low sensitivity speakers that you may want to buy later, i.e. you would be ready for great speakers like the Harbeth Super HL5+.

For a very sweet top end, Harbeth is probably the brand to go for. They are expensive, but fortunately not fussy about amplification. Their designer Alan Shaw argues that any properly designed and powerful enough mainstream amplifier will do, and for the forthcoming Bristol Audio show he intends to demonstrate their P3ESR mini monitor with a Yamaha AS 701. For his bigger speakers in a big room he recommends a lot more power (at the lowest price per watt).

OK, you seem to have reached a satisfactory solution for the small room and HT, using, as I uderstand it, the two smallest Polk speakers.
That leaves the music room question. The thing you and only you can decide is in which room you will wantto listen to music. Both are suitable, but the bigger of the two potential music rooms will allow even deeper bass to be reproduced without too much degradation. But in essence this is a lifestyle issue.
I can understand that you do not (yet) want to splash out on expensive speakers (I did not quite realize the Harbeth speakes are so expensive in the US, and I did not know exactly how much you could get back if you returned the bigger Polk speakers).
So, I assume you will want to keep one of the two pairs of Polk floor standing speaker pairs. In the bigger room I would opt for the biggest of them, in the mid size room I would try what you prefer.
And those speakers will need an amplifier. My advice would be to buy something that is not too expensive, but good enough to be used completely succesfully when you decide to upgrade the speakers. Fortunately, and contrary to what some here believe, good electronics do not need to be expensive.
For the mid size room I would recommend the Yamaha AS 801. It performs flawlessly (see the test results of their AS 700 that I posted earlier - essentially the same amplifier but wthout the modern digital inputs) and 2x100 watt will be enough to drive almost any good speaker in this mid size room. Harbeth’s Alan Shaw bought the almost identical AS701 for a demo room at the forthcoming Bristol show. Having seen the test results he did not even open the box and will not do so until the day of the show. So much for burning in or long listening tests to establish ’synergy’. He will not use fancy cables either. He is one of this world’s most respected speaker designers.
For the large room something more powerful would be required once you move up to more ambitious speakers (with your current quite sensitive Polks the AS 801 may still be just OK). But a room like this does indeed require big power, and that does not often come in integrateds. Hence my suggestion for something like that 2x350 watt Yamaha pro audio amplifier with either the Pioneer U-05 or the Oppo 205 as a front end, depending on whether you want to play discs or not. Both have balanced XLR connections for lower noise. One of these big powered combinations should be more than good enough to drive any high quality speaker (up to Harbeth’s top of the line M40.2) in a big room. I bought the 2x250 watt P2500S for my son’s birthday as he is planning to buy a pair of Harbeth M30.1s by next summer, and the resulting sound with the current speakers is as clean and precise/neutral as you could wish for. If you want to be absolutely sure, you can even go one size up from the P3500S, to the 2x500 watt P5000S. Harbeth recently demonstrated their big speaker in the Netherlands, and the power meter on the amplifier that they used indicated that at times it produced more than 500 watt on peaks of dynamic music. I just saw a P5000S on Amazon for $650. These excellent pro audio amplifiers can be so cheap because unlike audiophile stuff they are produced in huge numbers. Audiophiles do not want to know, and the cottage industry works hard to persuade everyone that somehow pro audio is no good, but proper tests show otherwise. Use these big amps with your Polks for now and you do have to be a bit careful, but pro audio amps have their own gain control and on the Yamaha’s you can set that behind a screw-on lid to prevent visitors (like my son’s friends) from destroying your gear. Similarly, they have adjustable high pass filters to keep the very lowest frequencies from wrecking your speakers.

Ah, you already own a decent usb etc DAC, at least that is what I understand the Focusrite to be (amongst a lot of other things)? Is that what I think it is? That makes it a lot cheaper, because all you perhaps need is an additional volume control in between the DAC and your great power amp (but from what I can see the Focusrite also has a volume control). With your beefy power amp I would suggest perhaps a basic analogue volume control as well, or for now perhaps some inline attenuators (plus a low setting of the gain controls on the power amp to avoid damage to your speakers): http://www.tcelectronic.com/level-pilot/
The only snag I can see is that your DAC has an rca output and the Level Pilot and the power amp have balanced xlr inputs. But there are cables to deal with that, and they are not even expensive (my son needed one too). Don’t get suckered into expensive cables.
All in all and for now I would not buy an extra DAC. I think you are done.

Look at the manual of the Focusrite. From what I can see it has multiple analogue outputs, so you should be able to connect both main speakers and subwoofer to the Focusrite. From what I can see the Focusrite has balanced analogue outputs with TRS Plugs. The Yamaha has both Neutrik and TRS inputs, so you should be fine with a TRS to TRS cable to connect Focusrite to Yamaha. This way you can use the high pass filter on the Yamaha to relieve the main speakers from the very lowest frequencies (the sub will deal with those). Set it at something like 40 Hz, or maybe a bit higher.
Consult Focusrite about how to connect the sub to the Focusrite. At the very least this needs a TRS to rca cable.
You may consider a DSpeaker Antimode 8033 room eq unit to equalize the subwoofer and tame room modes. See here: http://www.theabsolutesound.com/articles/dspeaker-anti-mode-8033-dsp-subwoofer-equalizer-tas-204/ I have been very pleased with mine. In your case the stereo model may be convenient because it takes L and R inputs and has mono outputs (ask the DSpeaker importer for which types of cables you need).
If the Focusrite somehow cannot output signals to both the power amp and the sub, you will have to connect at speaker level, using an attenuation cable from the power amp’s outputs to the sub. But it had better be able to handle the Yamaha’s big power.
Since the Focusrite does not have tone controls (unless hidden somewhere) and for as long as you are only using a computer as your source, install the free Equalizer Apo (plus Peace interface) software on your computer. This is an advanced software equalizer that you can use for basic tone control, but also for more detailed parametric equalization. In fact, if you use the REW software and a UMIK-1 calibrated microphone you can do subwoofer equalization with this as well. But REW is hard work. So to keep it simple, use the Antimode for sub eq and the Equalizer Apo (without REW) as basic tone control. Use the high pass filter on the Antimode to relieve the sub from havng to reproduce the very lowest frequencies (say, below 15 Hz). This is a large room for a modest sub, and lowest frequencies are the hardest work (they need much more energy).

Sorry for the confusion I created (it was too early in the morning and I was in haste): I meant to write amplifier but I wrote speakers. The output of the Focusrite should obviously go to the Yamaha.
I thought you were using a Windows PC.
The Yamaha is indeed a pro audio amplifier with balanced inputs (both Neutrik and TRS). That is advantageous because the connection will be less noisy if your pre amplifier/DAC has balanced outputs as well (as would seem to be the case). In the case of the subwoofer connection, trs or Neutrik to rca cables are readily available for little money and are unproblematic. The presence of gain controls on the amplifier is similarly nice, because it allows you to tame the amplifier’s pretty massive output.
As is shown in the link that I posted earlier, its smaller sister the P3500S was recently tested and measured exceptionally well. The P2500S that I gave my son for his birthday sounds perfectly fine, and I am sure measures similarly well. These Yamaha amplifiers have variable speed fans (unlike much of the pro audio competition), but they run so cool that under domestic conditions these will never come on.

I would use the bigger of the two subs in that large room. For connections, first investigate the analogue output types of the Focusrite (I could not quickly find the information). What are yo doing with the smaller Polk sub? Will you return it?
I think you are almost there with the music system for the big room.

Ok that is more info on the Focusrite. My only remaining question would be if you can play from two analogue outputs at the same time, but that is easy enough to test with your existing gear.
Continuing on the subwoofer side, I would return the Polk sub. It only creates problems in your small room, and it is not ideal as a second sub in your music system. For that music system I would strongly recommend the Antimode 8033 room eq system. It is cheap and does a great job without any fuss. For even better bass you could buy a second identical Klipsch. A second sub gives more power and a smoother response over a wider listening area.
That big room will almost certainly need some damping of high frequency reflections by curtains, rugs etc. I don’t think bass traps and the like are needed. The room is so big that room modes will be at pretty low frequencies where bass traps would need to be pretty enormous to have any effect. I would first opt for the Antimode 8033, and if you want an even smoother response over a larger area, go for a second sub. For the rationale behind mutiple subs, see here: https://www.google.nl/search?q=welti+geddes+multiple+subs&ie=utf-8&oe=utf-8&client=firef...
You have the space for what is probably the ultimate solution, the Audiokinses Swarm system: http://www.theabsolutesound.com/articles/audiokinesis-swarm-subwoofer-system/
This too can still be optimised by the Antimode.

Yes it does. However, the Focusrite outputs are not rca but TRS (balanced), and they are stereo. So you need a TRS (balanced) to rca (unbalanced) cable. If you are not using an Antimode in between the Focusrite and the sub, that cable should also be in a Y configuration, because you have to end up in one (mono) sub. If you were to use the Antimode 8033 Sii it has stereo rca inputs and mono rca output.

So your sub has L and R inputs. That simplifies it a bit. All you have to do is to run a cable from L out from the Focusrite to L in on the sub. Ditto with R. I am not sure about the plug that you linked to (call Focusrite for advice). See here for more information: https://www.presonus.com/learn/technical-articles/Balanced-Unbalanced
Try the Antimode whenever you feel like it - I think you will love it.
I would cross over rather lower, like 40 Hz for a start, but you can experiment.
The Yamaha AS801 will be a bit easier, but the sound should be virtually the same, with a bit less dynamic headroom in that large room.
Bi-amping is not worth it in your situation, and adds a lot of worries.
An Oppo probably only makes sense if you want to play discs and/or want to connect a TV screen.
Once you have this up and running, stop worrying about the electronics (they will be more than good enough, even with far better speakers). Your three weakest links will be the main speakers (start saving for something like the bigger Harbeths), the room modes generated by the sub (get an Antimode), and the higher frequency reflections because of the hard walls, glass windows etc. (get some curtains, carpets, or whatever).