Can the follower hold the 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.

Not only is the horse faster (not even considering their higher base stamina -- Horses can sprint much longer than a stamina-light Dohvakiin!), but they can be ridden while encumbered.

So if two Dovahkiin (Dovahki?) were to race across a flat plain, one on a horse and the other on foot, the horse-bound hero would win every time. Horses are marginally faster

So what about when you're not traveling in a straight line?

Turns out, the horse will still win, as, despite their cumbersome turning, horses flaunt a blatant disregard for physics that can quickly turn into an incredible timesaving path.

Consider: When you're on foot, you must run around any mountains in your path.

Horses, on the other hand, have no qualms with running straight up the mountainous slopes.

Image credit to Emperorr of Reddit.