Best Headphone system under $1000???

I'm not sure if it's reasonable to expect to "replicate" what you have in a headphone system, the experience of listening to music through speakers and headphones is very different.  Your budget allows for some decent components. 

If you're willing to buy used, consider the Audeze LCD-2.  They are detailed, but on the warm side and have good bass and will sound great with the types of music you enjoy.  You can get some new earpads for a set of used of headphones if you're squeamish about buying used.  The Audeze ones are about $100, there are some nice aftermarket ones out there for less.

If you prefer new, check out the Hifiman HE-560.  They are similar to the LCD-2, but are harder to drive and will require a more powerful headphone amp.  Another great set of headphones is the Focal Elear (or Elex from Massdrop).  These are more "neutral" than the other two I've mentioned, but are very detailed and easy to listen to.

An outstanding headphone value that I recommend looking at is the Massdrop Sennheiser HD-6XX headphones.  These are super comfortable and sound great and are only $200.

Like speakers, the headphone is going to impart more to the sound signature than the amp, but proper amplification is important.  I don't have any suggestions in your budget range, most of my amps cost more than that.  If you can find a used  Musical Fidelity M1-HPA or M1-HPAP, that's an inexpensive amp that I have owned that has a lot of power and sounds good.

The system I use and can therefore recommend is Focal Elears and Schiit Jotunheim.

I was going to mention Schiit, but I'm not familiar with the models that would fit in the OP's budget.  I have a Mjolnir 1 and it's a great amp.  If the Jotunheim is close in terms of sound quality and performance, that's definitely a good option to consider.

I agree with the comparison between the Elex and the HD-6XX, but would not call the HD-6XX boring.  I guess one man's smooth and easy to listen to is another man's boring. The Elex is definitely better, but the HD-6XX is a lot of headphone for the money. What I like most about the HD-6XX is how light and comfortable they are.

@james63 The HD-6XX are $199.

One of the great things about the HD-6XX is that since the HD-600/650 have been around forever there are a lot of aftermarket accessories for them - cables, earpads, etc.  Putting some nice earpads on mine made them a lot more comfortable.  I didn't notice a huge difference in sound quality as some claim, but the comfort level went up.  You may need to purchase a plate, depending on which earpads you get.  Check out modhouseaudio.com for those.  I don't remember which earpads I bought, but there are a number of choices out there including Dekoni, ZMF, Brainwavs, and more.

The amp in the previous post would be a good buy. I have the M1 HPAP and it's a solid performer. It's only good for connecting source components though, the USB is not very good. 

In regards to your question about cables, I have Focal Elex and Sony MDR-Z1R headphones, which both have the same connectors.  I've used a few different cables, mostly mid-grade to inexpensive cables. 

To me, there is not a noticeable difference between cables, which is also my experience with interconnects, speaker cables, etc. 

I buy cables for fit and finish, length, different connectors (balanced, single-ended), not for sound quality.  Which isn't to say that there aren't differences in sound quality, but that unless I A/B them back and forth repeatedly, there aren't differences noticeable enough (to me - your hearing may be more sensitive) to make me prefer one cable over another.  

How would you compare the Audeze LCD 2 against the Focal Elex headphones?

They are both excellent headphones.  The LCD 2 is going to have more bass and be a little more "rolled off" at the high end (the Audeze "house sound").  They still provide good detail and excel in the mid-range.  The Elex is going to be a little more neutral throughout the frequency range and I find them more comfortable to wear.  Which isn't to say that the Elex lack bass or that the LCD 2 is uncomfortable.  The Elex has good bass response and maybe a little more "realistic".  One of the things I like about Audeze is they come with really nice earpads and are comfortable from that standpoint, but they are a little on the heavy side.

I suggest checking out Headfi and Innerfidelity for in-depth reviews of both.