1) Yes. I do it all winter, using the inverter/battery combination and a smart charger/maintainer.
2) I leave mine plugged in all winter, just because I hate digging the shore power cable out from under the snow and ice. If I can find it under there (just kidding).
I'm using the factory installed PD9245 converter/charger plugged into shore power to keep the batteries up. The rooftop solar would probably maintain it by itself, but I'd rather plug it in, jic. It's not going anywhere when there's snow on the ground anyway, so why not leave it plugged in?
I'm across the river from SE Michigan, btw.
2018 (2017 Sprinter Cab Chassis) Navion24V + 2016 JKU (sold @ ????)
2016 Sunstar 26HE, V10, 3V, 6 Speed (sold @ 4600 miles)
2002 Roadtrek C190P (sold @ 315,000kms)