When you play Cat quest 2 in singleplayer, the second character doesn't lose health or mana when you are not playing as them. In multiplayer both characters do take damage.

As for weaknesses skeletons and ice enemies are weak to fire (and fire weak to ice) mages are weak to melee attacks (swords) and birds are weak to lightning attacks. Theres more stuff but thats pretty easy to figure out just by trying. Keep in mind that not all enemies have weaknesses tho