Do Hearthstones exist? If so, are there limitations?
Does your character have multiple set to bring them to preferred locations?

They do, Tev had one before he died, but it got lost in the shuffle after he was died and raised as a Death Knight. When used in actual plot, death gates seem to be way more flexible than “Opens a gate which the Death Knight can use to return to Ebon Hold.” so I figure that Tev primarily uses those for transportation over long distances, or whenever he needs to make a quick escape.

Does your character have only a few mounts they use regularly, or do they have vast stables filled with creatures they’ve collected?

The only two mounts that Tev has that are canon are his deathcharger, Retribution: 

And the Frostwyrm, Kragos:

When you dismount, where does your mount go?

Retribution is more like a summon, when Tev needs to ride him, he’s called from the Shadowlands, and when he’s done with him Tev sends him back there. It’s probably a bit more involved than the 1.5 second cast time that the game has, so Tev will keep him around if he may need him later.

With Kyragos, Tev had to raise him outright as a test of competency (back when he was with the Scourge), and while he’s sworn fealty to Tev, he’s an entity independent of him, and a member of the Ebon Blade in his own right. Kyragos probably just flies off to carry about his own business after he’s done with whatever it is that Tevruden needed.