I have yet to make a successful bow (no0b disclaimer), but I have been reading/working at this for 9mo or so and so far I only have mulberry to work with. Here's what I've learned about it:
NOTE - I only have red mulberry so what I've read pertains mainly to it specifically, though it might be the same for some of the other mulberry types.
1. Splitting sux a LOT when it's wet. If you can leave it somewhere (ends sealed, bark sprayed with pesticide) to dry for a year before splitting - morebetter. If you must split it now - pack a lunch and don't screw up your axe handles like I did.
2. It's a great wood for bows - very close to the mythical osage orange in most respects, but not quite as hearty so be a little more conservative with your design than you might osage.
3. Like osage, it's mainly good for its heartwood, but it is possible to make a bow with the sapwood (I think I saw someone on here that had both sap and heart in their's)
4. No need to heat temper the belly.
5. Chasing a ring is your best bet - almost identical to osage.
That's about all I know right now. I also just finished making a couple of axe handles with it as well (yes, the same ones I boogered up splitting my first mulberry log). It's not quite Osage, but it seems close enough (from what I've read/seen) that you can follow mostly the same approach for making a bow.