Can I lose a follower by horse riding over a long distance?

Followers DO NOT GET LOST. If you are too far away, they either teleport to you when you load up the next zone (eg. a trader's house) or they spawn somewhere close to you, where it only takes about a minute to catch up. If you find a safe place and idle there for a few minutes they mostly find their way back to you. [EDIT: Note that you cannot ALT+TAB out of the game, because it will freeze]

I once found that Lydia has grown tired of waiting so she returned to Dragonsreach and was sitting at the long dining table. Then you can ask her to follow you again. It's also possible she's still on her way back, that you arrived before she did. Waiting about a day or so is usually enough for a NPC to catch up to you.

Yes, you can. I can not say if permanently or not, though. My best guess is no.

I've just recently lost Lydia (her bed in my house is still "owned" though, maybe I should go look for her.. maybe she needs my help!), one of your first companions. When riding, she frequently started attacking random crabs and thus falling too far behind or her pathing was just so bad that it took her forever to find to me.

If you are really attached to your follower, you should play it out and wait for them. You can also tell them to wait for you somewhere, ride to your destination, then use fast travel to get them.

Lydia simply despawned for me several times when I rode away from her and upon changing the zone she re-appeared every time. But then just recently, she was gone entirely and I can not figure out where she is or if she's even still alive. But that's really my fault I guess.