I applied to the Bloomberg website in late July, got my first email to do a phone screen in late August. Bloomberg doesn't recruit at my school so I did not have any on-campus interviews.
Phone screen was relatively easy (see Question #1).
My on-site interviews had 4 rounds, around an hour each with 15 minute breaks in-between. Something I've heard is that every candidate gets two rounds but if they escort you out after two, you probably won't get the offer. I had two technical rounds, one with HR, and one with a manager.
The interview experience was really nice, especially the on-site and Discover Day. Bloomberg will fly you out and put you up in a really nice hotel. You get a black car to take you from airport to the hotel. They'll give you a tour of the office building (which is really really cool). You get a reimbursement card at the end for $100, but in my experience it wasn't enough to cover expenses.
Discover Day is a chance to interact with the other potential interns and full-timers (all have offers). You also get a chance to talk with Engineers, and even casually talk to several teams. I suspect this was a way to preliminarily team match.