Furnace running requires hookup or engine running?

The house battery is beyond the end of its useful life, it seems, from what you've written. I suspect it gets very rough usage as basically a rental RV, probably being fully or nearly fully discharged many times, so they probably don't last very long.

The generator does need a reasonably decent battery (and good solid connections to it) to start; it takes a lot of current to crank, compared with typical house loads.