Hi, and welcome to the forum.
1) Fin Rot: Yes, he might have some fin rot. This is probably due to high ammonia levels in his cup while he was at the store. The best way to treat fin rot is with lots of warm, clean water. If you can provide that, it should heal up. If it doesn't get better soon, let us know, and we'll give you additional advice.
2) Temperature: A temp of 78F is just fine.
3) Eating: A lot of new bettas won't eat for the first few days. Don't worry about it. Simply offer him a pellet. If he doesn't eat it, remove it so it doesn't foul the water. Then try to feed him later on. He should start to eat in a few days. If he doesn't, let us know.
4) Pooping: Yes, the thing hanging from his belly was probably poop. It's usually red-brown in color. (But can vary, depending on diet.)
5) Slimy stuff on fins: I do see some white in the picture. But is it slimy? Or fuzzy? Has it gotten better, worse, or is it staying the same.
6) Lethargy: How did you acclimate (introduce) him into the tank? It's a good idea to introduce them to their new tank slowly, so their systems aren't shocked by the different water chemistry.
7) Bubble nests: Bubble nests don't necessarily indicate 'happiness' or 'sadness.' They're built by male bettas in order to attract a female. Many fish also build them when storms are about to go through the area! They're pretty cool, aren't they?
8) Plastic and Silk plants: Yes, they can tear their fins on plastic plants. You can use soft ones though. If you have a pair of panty hose, pull it over the plant. If it snags the hose, there's a chance it could snag and tear his fins. Toss it..... If it doesn't snag, then it can be safe. It's your call on whether to use them or not, though!... Another option is silk plants. They're available at petstores and online.
OK, I think I've covered everything! Let us know whether the 'slime' on his fins is slimy or fuzzy. They mean different things. How is he doing tonight?