Some bandits will warn me to stay back when I approach them instead of attacking me on sight. It seems to be happening more often the longer I play. Is this just totally random, or does something cause them to give you a warning?

It's random. Some bandits are more territorial than others. Usually when you're approaching the entrance to one of their caves.

I think it also has something to do with what level you're at, doesn't it? At lower levels they figure they can easily take on a wimp like you while at later ones they consider it safer not to create a scene unless they absolutely have to.

This also reminds me of when you a thief tries to rob you, and you successfully intimidate them:

"Well, uh, I can see you are not one to be trifled with. Let this be a warning to you!"

Needless to say, many warnings have been met with a sword to the face.

I don't really think so. They just warn you if you're in they territory, and if you approach them closer they will attack you. And if you stay too long in your place, they will attack you too.

I like it when my character, wearing Ebony armor, with Ebony weapons, an expensive circlet and a follower with basically the same kind of equipment, persuades the thief that she hasn't got any money.

Well, of course, she might not have any actual MONEY as far as the thief knows, only very expensive gear...

They don't mind if you point an arrow or spell at their face for some reason

Seems like they don't bother me when I ride around on shadowmere. They don't make a statement or anything, like when I pass Valtheim towers, the chick that's there walks at me but doesn't start and dialogue as I go by.

Hardly any bandits on my game since I clear almost every cave, settlement etc that has bandits (until they respawn of course) when I'm trying to run the ring of Skyrim as a Werewolf. I eat so many people it takes a literal game week to change back. .