I interviewed with the New York Fed and got an offer (but turned it down), so I'll try and answer what I can. Keep in mind this is information for the RA job after college, but I think a lot of it will translate over to summer internships. Hopefully someone else can fill in the gaps:
1) Jobs are fairly competitive. I think people here tend to overestimate just how competitive, but I come from a top university and all of the RAs that I spoke with came from top 15-20 programs or top LACs. I don't know if I saw anyone who was "top 100" -- but this was for the New York fed full time offer, which I believe is the most competitive. It might not be as hard to get an internship at, say, Philly or Boston. I did get rejected by the Richmond fed the summer after my freshman year, but they strongly indicated that it was because of my class year, and told me that I would get it if I applied again after sophomore year.
2) Yeah an internship at the Fed will look good. Any research looks fine. I spent the summer after my sophomore year at an economic think tank and people seemed to like that, and that isn't as good as one of the better feds. I'm not sure exactly how good, but fed economists are generally on par with a top 50 or so program (depends on branch), and some are really good.
3) Yeah apply widely. Internships can be a crapshoot, so apply to a lot.
4) Depends on the prestige of your home institution, really. I was actually in a position where I was better off RAing for professors at my university rather than at the Fed. But since you're at a top 100 university, it seems like your econ department might not be all that strong. In that case, I think the fed would be better.
One thing that I've stressed before is that the Fed branches are generally in cool cities (Philly, Boston, NYC, DC, etc.). So even if it is marginally better to stay at your home university, that might be boring and your summer might not be that fun outside of work. I had so much more fun working in DC than staying in the small city where my school is located. Probably one of the funnest summers of my life. Another thing to keep in mind.