Once you interact and jump on a horse, not just on a farm, but anywhere in the world of V Rising, it's yours. I'll expand more on that in the next section.
You can't buy one or breed one. You can initially only find it in the Dunley Farmlands region, but after some of them are found and taken, you will probably find some horses during your travels, even in Farbane Woods.
Well, once you get on a horse, the next time you press the "Tab" key, you will see the horse menu next to your gear window even after dismounting it. There's a tiny note icon on the top right corner of that menu. Press on that, and you will be able to change your horse's name.
Once you do that, you probably think that it's yours forever. But that's still not the case. If you leave your horse out in the open, another player can take it, and that's it. So it would be his horse until the next player takes it. That's just the way it is in V Rising.
Your best option to keep your horse while you are playing or turning offline is to keep it in your castle's territory, where it can't go anywhere and where other players can't reach it (well, unless you are being raided).
Sort of, yes. If you remember, you now have a horse menu. In it, you will see that the horse needs water to live. Obviously, like your Castle Heart, you will need a specific resource while offline to keep things operational. So, you will need to find a 'Water-filled Canteen' for the horse. Each one will be enough for 7.5 hours.
If you can't find a filled canteen, you can create an 'Empty Canteen' in your Tannery or in your personal crafting menu (Tab), then find a water source to fill it up. You'll need 20 Plant Fiber and 4 Leather to create the canteen.
That's it really, the horses are the fastest and will be great for your travels during the night and during the day. You can also use the horse to attack while mounted, and some weapons will act differently while mounted. For example, the Reaper does a more significant swing than usual, which is excellent for farming large areas.