Eads Wash was actually pretty neat. There is a very simple bathroom with a men's and women's side. The final stretch of road on the way to Eads is dirt and did have a few rocky patches (nothing I could imagine needing 4 wheel drive for). However, every side road we took looking for a campsite had spots where there was powder soft dirt that could easily end poorly without 4 wheel drive. During our stay we did have to watch another group of campers dig out their truck/trailer less than 50 feet from our campsite. As such, my suggestion is not to plan on camping or going off the main road without 4 wheel drive.
The "campsites" aren't clearly marked but you can find plenty of places with dirt roads and cutouts big enough for a tent or trailer. Most didn't have very established fire rings so plan on rock hunting and making your own. There were multiple places you could get right up next to the river. There are signs up warning of flash flooding though so be aware of the weather that effects the river.
Eads Wash had very few cacti in the area we stayed in. It is obvious this area floods so the plants largely resembled bamboo and greasewoods and there were not a lot of large trees. However, this seemed to help with the wood situation because we were able to find piles of dead/down wood fairly easily within a few miles. We camped about 100 feet from the river but were able to easily hear the water running all night. Being in a wash did effect the temperature a lot. We saw freezing temperatures at night (long enough to form ice in our jugs of water) and low 80s during the day (again, very little tree shade to get out of the sun).