> a: Will the AC unit run off off just the batteries?
No, but the heat will.
> b: What size of generator would be best for the above scenario? I see a lot of conflicting info on this? The description for the Yamaha 2400 says it will run a 13500 AC however other info I found said you need at least 3000W of starting power (vs the 2400 starting of the Yamaha linked) https://www.amazon.com/Yamaha-EF2400iSHC-Starting-Portable-Inverter/dp/B00AYUWDZG
I have a 4000W Powerhouse Inverter generator, and it would stall out starting the generator, until I installed an Easy Start from MicroAir. Now, it never does. I would recommend getting the Easy Start and then you should have no problem with that Yamaha.
> c: The Nash 25c has a generator enclosure that's vented and plumbed for propane, would it be dumb to not get a propane genny (or even gas) that fits in the enclosure as that sounds nice and it's locked up when in use (may go skiing or dirtbiking and leave the trailer while the generator runs and fills up the batteries).
I have no experience with this, so I will have to let others answer.
> d: Any idea how long it would take to fill the batteries (sorry don't have specifics no the "RV" batteries that are being included but) a general idea would great. Like 5-8 hours or just a hour or two?
Depending on size, it will probably take a couple of hours to charge. TBH though, my batteries last a long time (I have two deep cycle batteries) when just using them for lighting and incidental other things.
The Hondas are always a choice, but pricey.
This thing is fantastic (also pricey) https://www.amazon.com/Yamaha-EF2400iSHC-Starting-Portable-Inverter/dp/B00AYUWDZG