Do NPC Companions level up with you even when they aren't in your company?

^

Approve.

If there must be some mechanic where companions fall behind in terms of XP, maybe make it that you can take them to an arena or trainer where you can pay gold to catch them up to the PCs level quickly. That way there is a cost barrier, but it still allows you to pick companions without having to worry about them being woefully underlevelled.

Which will only raise the question...

"If you have an awesome trainer... why not pay him to get you to [max level] right at the start?"

So, no, don't think that works.

There are several ways the game can "punish" the player for not keeping a follower in the party. For example, they can have the follower "follow" their own path of level up progression, rather than the one you choose.

Well, there is a big difference.

Having them lack a level or few still allows catch-up. It takes work, but it can be overcome.

An auto-leveled teammember however can become permanently useless, and there's nothing you can do about that.

That's not good. I am all for making people think before they do, but ruining their game for the heck of it, no thanks.