Dexx,
You have asked a lot more than the three question you think you asked and answers are more than a little complex.
Vocabulary for today:
A Converter make 12VDC from 120VAC.
A Converter/Charger makes 12VDC from 120VAC, but has the control to safely recharge the house bank.
An Inverter makes 120VAC from 12VDC.
For #1, Yes, you can put in a mini-reefer and use a serious inverter to run it from a bank. It will take a serious bank to run it more than a day. The starting load of the Chinese compressor will require a larger inverter than you might think. The power consumption is also always great than published. Your 60AH/D is probably very optimistic. Then there is the fact that the condenser is the sides of the unit and must be in free air to cool. I don't what jars you think are 155AH, but you have to remember that you can only use about an half of that before the jar is damaged. If you are thinking of single 12V jars and putting them in parallel, think again. Lead acid (LA) batteries do not play nicely in parallel. You would be much better off with 220ah golfcart jars in pairs in series.
You say broken beyond repair. Did you investigate any of the rebuilt cooling units? Usually about an half of the cost of a new. On DC or AC, they eat power too. But on propane they run forever.
So Much For Reefers for now.
For your second, without any information about the equipment, my best guess is that you have an antique charger/converter that is switching out for some reason. These were old uncontrolled brutes that had both a 12V supply section and an battery charge section because they had nothing like the control required to change a bank of jars and not fry ever lightbulb. It sounds like is it deciding that the AC supply has faulted when you shut down the A/C and so puts the house back on the house bank. Replace that with a good 3 or 4 stage smart charger. (Look at Iota or Progressive Dynamics.)
About the third, Both shore power and the APU (genset-generator) supply the unit with 120VAC for the system that uses it. It is rare that this device also charges anything. Typically, there is a Converter/Charger that is powered by what ever AC is sees and it charges the house bank and supplies power to the 12VDC system.
The main engine start battery should be completely isolated from the above unless someone has added a specific charger for that.
If this is not a complete answer the questions posed, try again.
Matt
