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

It seems that my follower likes to get very far behind when I go and ride a horse over a long distance, can I lose him by doing so or will he manage to find his way to me? Can this result in his death because he enters a strong battle? If this can cause me to lose him, can I somehow give him a horse?

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.

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.

Usually Lydia gets stuck in a house (I would assume other Followers do as well)

Recently I had the worst experience. I lockpicked a house and somehow SHE got stuck saying "You shouldn't be in here!")

Well I left, and played for a while without her because I figured she would eventually show up.

Nope. Eventually I had to go to the house. After some weird combinations of using "Voice of the Emperor" (Imperial perk) I was able to calm her and talk to her. This didn't work every time and I still got the guards called on me.

Another time she just walked out when I did. Most of the time though.. I would walk away and she would just stand there or do her own thing and just go "Hrm?" when I talked to her. I tried telling her to go out the door by issuing a command and she just said "I can't do that."

What I ended up doing though was attacking her a few times. This got her mad and she started to attack me. I then ran away from her and fast traveled back to Whiterun and she was at her desk by the Jarl. She talked to me and funnily enough said "Yes my Thane?" and I hit tab and she said "Get away from me!" she continued to end every thing with "Get away from me!" for a while until that eventually stopped too.

I got her back though, with some creativity, and she's nice and happy again...