1: How would a Skyhammer handle various off-pad aborts?
I sized the engines such that you can lose one right off the pad and still be OK (thrust-to-weight still greater than one). The lack of any solid rocket motors should open up the window of doing powered aborts significantly with respect to the Shuttle. All the internal fuel should help with this as well.
With that said, I think the fun of Orbiter is in going to space rather than not going to space, so worrying about aborts is pretty far down the list.

Once I have the Skyhammer flying I would love for people to figure out how to fly it in aborts though. That would be awesome.
...how does bailing out work?
I haven't placed the crew hatches yet, but this will definitely be a consideration. In general it will probably work like the Shuttle. The caveat above applies, actually implementing any abort logic is pretty far down the list.
2: Just what are the landing mass limits for Skyhammer? If it's like the Shuttles, it's probably less than it's max payload.
Is that true? I'm not a Shuttle operations expert by any means, but I doubt they would be allowed to fly a payload they couldn't land with if they had to abort. The only exception I could think of would be the Centaur-G which they could dump the propellant from, but even then I think it was just an explosion risk (and not coincidentally it never flew). Open to being corrected on this, though.
As for Skyhammer, I would expect it to be able to land (gently) with zero fuel and max payload.
3: Would the Skyhammer be able to handle propellents in the payload bay, like for upper stages or delivering that propellent to a depot?
Absolutely! I would really like to have an integrated propellant transfer system with Skyhammer, so you can assemble things in orbit, service/fuel them, and go exploring. Hooking up propellant fill/drain lines to the payload bay is no problem. You have to worry about propellant dumping, but you probably have to dump the orbiter's internal propellant before a landing (abort or no) anyway.