1. Is running an ethernet wire from my streamer/DAC to the main pod equivalent to direct wiring to the old stand alone modem/router?
If the main pod is hardwired to the modem, then yes, it's basically the same as direct wire. However, if you can, try to wire it directly to the modem. Direct to the modem is the best.
2. If I move the modem and main pod; can I run a cable from the main pod to an ethernet port ( to be installed ) in the cabinet and plug in my streamer? Will this be nearly equivalent to hard wiring into my old modem/router?
Yes, basically the same as #1.
2 things you should do.....
1. Make sure you use high quality certified C6 cables. Don't need to be fancy, just make sure they are certified. This really makes a difference. I get most of mine from work, but BlueJeans has good cheap certified cables.
2. Try to have the least number of "hops" from the DAC to the modem. Each hop can/will add jitter, noise, can have lost packets. Most home "mesh" networks use cheap components are mostly crap for anything high end. Hardwire will always beat a mesh network.
If this was my network, the provided hardware would be given back, then purchase a good modem, hook that modem up to a little router, with a wi-fi point plugged into that. Anything that needs good fast signal, have it plugged into that switch. If you have a quality wi-fi access point, it can almost be as good as wired in places you can't run a wire.