Oriah: I didn't see the river after the storms because I don't live near it now. However, when I was younger I lived right on the river in downtown Chattanooga, and after heavy rain there are really strong currents out in the channel. It's a heavily traveled and heavily fished river though, and most of the time if someone goes in, the body is eventually found.
Caves: Most caves in this area are limestone sinkhole type caves that go straight down. I would say the vast majority of openings are far away from roads in rocky terrain, but these old mines might have openings close to one of the dirt roads. Could you drive right into one of them? Probably not, but you'd have to go out there and inspect each one individually, imo.
It does appear that several of the mines have road access by 4 wheel drive. I did a quick search on the collapsed vs non/collapsed shafts etc. I believe you are right in that each would have to be checked individually.
As for the river- I'm looking at the water levels and currents right now. It's suprising how far a very strong current can move even very heavy objects. Also, given the nature of the storms, I would think there was quite a bit of debris in the river afterwards?