Also, is it possible to level them up? Or do they level up just like me by fighting and increasing their skills?
They are locked at the level on which you hired them, and will not continue to level up as you do.
To set them on your current level(should be higher) you can:
This resets their stats to the current player level, and also does not appear to reset their inventory or quest scripting (although you will need to take an item from their inventory and put it back to get them to re-equip any non-default equipment you've given them). However, it has no effect if they are currently in your party. Essential characters can not be transformed, therefore essential followers can not reset their stats unless using console commands.
There are a couple very specific occasions in which certain Followers will level up due to scripted events in quests.
Note that most of followers can reach max lvl 30 so updating them past this level would be useless.
For those on PC check out
The UFO - Ultimate Follower Overhaul mod allows you to view your follower's level, specific skill levels and other stats (through a conversation option):