1. There is a battery house/coach disconnect switch (aka "salesman's switch") that should be used to disconnect your house batteries any time you're in storage. Depending on your batteries' condition, if left connected, they can easily discharge in a couple of weeks due to a number of "parasitic" loads such as your propane monitor, etc. Bad batteries can discharge even if disconnected.
See page 6-8 of your operator manual for info on the disconnect switch:
2. I looked at the chassis wiring diagram for your rig and it appears to me that your generator starts off of your chassis battery:
3. If your house batteries are original, without question, they should be replaced (4 - 5 years is typical). You can replace them with two similar (or if they'll fit, larger) "marine" batteries but, IMHO, a better solution would be to replace them with two 6v golf cart batteries connected in series to provide 12v. The typical "marine" batteries found in most RVs are a hybrid starting/deep cycle batteries and aren't as robust as true deep cycle batteries. Golf cart batteries are true deep cycle batteries and will give you more amp hours and last longer. They are taller than your existing batteries so measure your available space carefully. You may have to modify the hold-downs. There are "true deep cycle" marine batteries available but they're generally pretty expensive compared to golf cart batteries.
Trojan golf cart batteries are generally acknowledged to be high quality but I and many others have had good results with Costco, Sam's Club or Walmart versions at a lesser price.
There are many sites that will explain how to replace two 12v batteries in parallel with two 6v batteries in series.
Regardless of which way you choose to go, keeping the battery fluids topped up is essential for good battery life.
4. It doesn't get too cold here in the Sacramento area so I can't address the winter issue.