Invertor, how do I know when batteries are getting low?

Yes the battery can be damaged if you discharge them to much, and don't recharge them soon after using the power. A couple of days is fine, but a week can do a lot of damage to the battery.

You should have a battery meter, that says "full 3/4 1/2 and 1/4 or it might say Full 2/3 and 1/3. It is best to recharge the battery very day for a hour or so, unless you also have a solar system that will keep the battery full during the day.

Fred.