A warm moist place is best for them and the labyrinth organ they use to get oxygen from the air, hence why he comes to the surface. If you are concerned about jumping you can place a CD case over the top of the bowl, a piece of paper, etc. Since he is ill he probably won't be jumping anywhere. Good luck! Just keep up water changes, regular water from your tap plus dechlorinator is fine, you don't need to buy special water for him.
Marbles are ok, yes. The way I clean my smaller tanks is by putting the betta in a cup and then letting hot water run through the tank and gravel, swishing it around until there is no more uneaten food or poop left in, and then restock it.
If your plastic plant is sharp enough to snag pantyhose its sharp enough to tear his fins and should be removed. The type of gravel you have can also be sharp sometimes and damage their fins or scales so look for that, marbles are round and might be better.