Companions will not initiate a fight unless you attack first. They will not blow your cover when playing stealth. If you attack an enemy (or you are discovered), companions will back you up and attack anyone who is hostile towards you.
You can tell your companion through dialog how you want them to fight. The options are:
General - Ranged, Switch to Melee if enemy gets too close
Melee Only - If you are fighting vampires, most ranged weapons are useless.
Hide - (Don't fight)
Companions use a different system than the PC for health. They can take a certain number of hits based on their endurance and your level. After they absorb their max hits, they fall to the ground unconscious for 2 min or until combat is over (whichever comes first). When they get up, their hits reset. These hit counters do not reset between fights or even areas. They only reset when a companion falls and gets back up.
Your Level is based on your attributes, skills and disciplines. Following the tutorial, most people will start the game at level 2. If you max out all of your attributes, skills and disciplines, you would be considered level 20. Note that unallocated experience points do not contribute to your level, thus companions will remain weak in combat if you stock pile your experience.
As a rule of thumb, most companions can take a number of hits equivalent to your level. (Give or take up to 20 percent based on their unique stamina)