1. My local Airstream dealer has the gasket sets in stock, and any good RV dealer should be able to obtain them, and post them to you. You should ask for the full set of gaskets required for a routine overhaul. When you take the furnace to pieces, you will damage or destroy several smaller gaskets, and these will need to be replaced. One example is the small gasket where the burner tube enters the burner. A competent dealer will have a shop technician who will know which gaskets are required.
2. Yes, you need to separate the burner and exhaust and inlet tubes from the remainder of the furnace in order to clean them, and yes, you will need fresh gaskets available, as the originals will probably be damaged on dismantling.
3. Yes, any perforations in the tubes or burner will lead to a dangerous leak of exhaust fumes into the trailer. Scrape around with a long screwdriver after blowing them out with compressed air, to see if any corrosion is significant.
4. I'm only guessing here, but corrosion under the pan could be caused by pooling of condensation drips from the burner assembly. If the burner is cold from not having been used for a while, and warm moist air from the daytime atmosphere hits the cold metal, condensation will form. I spent several years worrying why I had an untraceable leak in my sailing yacht. I eventually discovered it was pooled condensation from the outside of a diesel fuel tank.
Good luck with the overhaul. It's a good opportunity to tidy up the wiring behind the furnace, wrapping it in tape to make a neat loom. Also, you could take the opportunity to install some wiring for LED warning lights on the outside light switches by the entrance door.....etc etc.....