3...Bc5 is a very normal move, played by many GMs.
In particular this sort of game is often reached via anti-Berlin move orders e.g. 3...Nf6 4.d3 Bc5
As for how to conceptualize the move 6...Bb4+ and 7.Kf1 the idea is (trying to keep it simple here) white has two threats... so black plays a check to try and gain time to wiggle out of it... but the bishop on b4 is trapped / not safe, so it doesn't work.
When your opponent has multiple threats playing checks, captures, and counter threats is the way to go, but in this particular position it doesn't work out for black.
1. e4 e5 2. Nf3 Nc6 3. Bb5 Bc5 4. c3 d6 5. d4 exd4 6. cxd4 Bb4+ 7. Kf1!! and black loses his dark-squared bishop.
I'm very surprised how often this comes up in my games and unless you really know your theory 7. Nc3 seems so much more natural for white and is what I played until I really started studying my games with a computer engine and saw this beauty.
Is there a name for this? Have you all seen this much? Obviously black has not not know a lot of theory because 3... Bc5 is not a common move although it looks pretty natural developing the bishop outside the pawn chain.