I havnt gotten any wrts in many years, but models that you could
re-flash were pretty limited. It maybe that there are plenty of
flashable models now.
You can look up models here, and find out details, hardware and
software vary a lot.
https://en.wikipedia.org/wiki/Linksys_WRT54G_series
The WRT runs off 12v, and I've run them off old car batteries for like
a month at a time. I think the board really runs 3.3v, and you could
probably rig them up to run on a few small batteries if you are
clever.
Im pretty sure many firmwares support WDS, I think I've even mixed
firmwares in a WDS setup.
What firmware are the cool kids using these days? openWRT? tomato? DDwrt?