Hello! I'm not a pure IFer, but follow some basic guidelines from it that have helped. I'll do my best to answer what I can, and maybe someone with more knowledge can come correct me if I'm wrong
1) 1300 calories should be fine! I'd say as long as you're still in a deficit, you shouldn't have any problem with this one. IF is designed to give you a more satisfied full feeling when you do eat, instead of frequent grazing which doesn't really give that "oh wow I'm full" feeling. You're eating the same amount of calories that someone else would eat over the course of a longer period of time.
2) I've never used the bulletproof coffee (actually I hadn't heard about it till now, and had to google it lol!), so I don't think it's 100% necessary. Regular coffee does have a slight appetite suppressant, so if you wanted to hold off your hunger for a little while longer, I'd imagine that's a good way to use it. This questions is probably better for someone who knows more about the IF system then I do
To me, it just kinda sounds like a marketing scheme, but I could be wrong!
3) This diet can be for anyone! If you like to experience that full feeling when you eat, I think this would be good...same if you just don't like that almost-but-not-quite-full feeling of eating multiple times throughout the day over a longer period of time. The issue that some people run into is the hungry feeling...you will feel it, and it can be uncomfortable at the start. Drinking water helps (and like the previous question, coffee can help too, but in moderation). But the uncomfortable feeling does go away, and after a while you realize that it's okay to sometimes feel a little hungry now and then (I've noticed it helped me recognize thirst more).
Unless you ate multiple thousands of calories over what you're supposed to, I doubt what you gained is fat, and is probably just water retention or hormones. You'll probably see it come off in no time! Hope this helps!