If you drive your car every day, then an extra jump start or 2 per day shouldn’t be very hard on your battery. But if they’re using it multiple times in between times that you drive it, then yes it can draw it down to the point of damaging it, and then you’ll both need new batteries.

If the coolant level was at the minimum, then it wasn’t too low to cool the engine. The problem is more likely to be a stuck-closed thermostat, or maybe a failed coolant pump.