electrical system

If you're conected to AC your converter should supply your DC needs as well as charge your batteries.

Depends what is meant by "weak". A "low" battery will simply get charged up, no problem. A battery with a bad cell could overwork the converter and cause a problem.

Land line.. there's a new one for shore power.

You can completely disconnect your batteries with no harm.

You camper is literally "plug and go and forget it." You should never have to alter anything on the electrical system. The converter on your camper knows when and how to charge everything needed and divide the DC power to the battery when it needs charging and still divert DC power to everything inside your camper that runs on DC. Literally, plug into shore power and go. Then forget about it.

Some older converters require a 12 volt battery to "smooth" the voltage. Give how poorly many converters work (even new ones), leaving connected is my default choice.

Batteries fail in 3 ways
High Resistance (Will have no effect on the converter)
Low capacity (likewise)
Shorted cell (Problem may appear).

Monitor charging current ... Normally if the batteries have been run down a bit you will have fairly high charge current when you first plug in.

But as the batteries fill the charge current drops. Less than 1 amp when full.

A Clamp on Multi-Meter with DC capability (not all can do DC) will indicate the amount of current flowing.. SOME may even do direction but mine does not.

Battery will last longer and stay stronger if kept on charge and full of water.

2X...Dutchman sport.