1 Most popups have a house battery as standard equipment. I suspect the previous owner removed it for whatever reason. Therefore there should be some connections for a battery on the 12 volt
converter. On popups I've usually seen the battery mounted in a plastic box on the tongue, just behind the propane tank.
2 I'd do some research and see what capacity the original tank was and go with that size. Anything larger and you effect weight distribution, existing structural, etc. I'd assume there is a check valve to prevent the tank from back filling but if not that would be easy to install.
3 I'd need to see a photo of the broken piece but really most RV drain stuff is the same as residential and you should be able to find what you need at Lowe's or an RV parts dealer.
4 Now this is the rough one. I know that AC units are a common option on popups these days. What is not widely known though is that with the AC unit, stronger lift cables are installed for the roof. You will need to look into that first, that is if the existing cables are strong enough or how much it will cost to have new ones put in, or how tough a DIY job it is.
For popups with an AC unit, the power cord for it is usually sewn into a sleeve or covered by a velcro flap, in the canvas in the rear of the driver's side, next to the lifting column. Easy enough to check for that. Whenever I have seen vent openings in roofs that are pre-wired for AC, there is usually a covered electrical box next to one side of the vent, clearly labled that is what it is for.
If you don't have the wiring in place to an existing roof vent, I seriously doubt that the lift cables or the roof structure can support one. Since I really suggest you contact Palamino about this, please post an update on what you learn.