I agree. Microwave is a 120 volt side of life device
Everything else you describe as not working properly is teh 12 volt side
Shorted cell in battery would be my suspiscion #1.. Converter issues #2
Step one: In battery compartment mark all wires hooked to the POSITIVE terminal with red paint or tape, remove and tie them all together, insulate (Plastic wrap is good for this)
Mark all NEGATIVE terminal wires with BLACK remove, insulation optional.
Remove battery or batteries to bench for external charging with a quality 3-stage smart charger
Remove to automotive store for quality testing.
IF You find you need to replace.
A GC-2 Six Volt Golf Car battery fits in the tray for a Group 24, only it is taller so you will need to make some changes (Longer bolts on the hold down clamps may be needed)
You wire them up like this
Negative side {6 Volt Battery} +Jumper- {6 Volt battery} Positive 12 volts.
They do drink Distilled water from time to time but they reward your effort with LONG LIFE.