I know it's a shared trait, but this character can't even heal 3 clicks

Actually, it could potentially benefit from healing 3. On its last click if it kills a codex fig on a push, then it would take 1 damage and heal 3 putting it top dial.

Is that how it works? I thought because pushing was unavoidable damage that it would be dealt immediately and not after the ability activates.

If the dial turns to a KO click, it is KO'd and can't be healed. After resolutions, you immediately give the action token, which immediately deals pushing damage, which immediately KOs them. Healing after resolutions takes place after all the "immediately" effects.