well, it's possible in RL, just not convenient, and takes a bit of practice to make even bad casts. I've actually seen people flyfishing from horseback on outdoors-oriented TV shows.
The real questionable "on horseback" things in UO are
Lumberjacking (where's the leverage?).
Crafting in General (oh, yeah, I want to see someone sew a straight line on a horse, or use a forge and anvil with the sparks flying, or even saw while on a real horse - and don't even get me started on cooking, inscription, masonry, glassblowing, and alchemy). At best, bowcraft and some aspects of Tinkering might be possible while mounted.
You can't MINE while on horseback in the game (but I think you can smelt the ore while mounted), you can DIG UP A TREASURE MAP CHEST while mounted.
A horse should be too skittish for a vampire to mount.
And, I'm just sure that the drake or WW you're taming can't smell that DINNER you're riding.