First, welcome to the club! Argosy's are wonderful trailers, I'm sure you'll get a lot of fun use out of it. Wow, $577 can go to a lot of repair, I'd hold off if I were you. I really didn't get too much out our inspection when I got it, but learned much more from others here and at forum rally's. Some thoughts
1) Axle - probably should consider it "needing replacement soon" if it's at all original. It probably has frozen in place by now because of age.. you can go to Inland RV's site and see how to inspect your axle angle.. you want the little arms pointing down. That said, they may be pointing down, but frozen in place. But DON"T be scared. You can still use your trailer, just keep in mind that it will ride slightly firmer. The shell likes a soft ride, but honestly, I have TWO axles (well 3 if you count my '63) to replace, and I STILL have to replace the ones on the Argosy (we've used it for 6 years this way).
2) Bearings.. just have them repacked. You can go to ANY trailer service place.. heck, try finding a farm shop in Texas, it'll probably be cheaper than in the city. You might want to search the treads to see who in TX like the service they got and from whom.
3) The fridge.. if you don't smell ammonia, and don't see yellow powder on the back coils from the access panel on the outside of the trailer, plug the trailer to 110 V and see if it gets cold OVERNIGHT. It won't cool as fast as your home fridge. Then fire it up on propane and see if that works.
4) 30 amp.. if you want your AC, then you want a 30 amp. Your pigtail probably has an adapter on it if it doesn't look like a large three bladed dryer plug (which it ISN"T so don't plug it in there). You can pull off the adapter and see the 30 amp plug. Go ahead and plug in to 15amp to use your other stuff though, and see if the converter is working to charge the battery.
5) You might want to see in the "willing inspectors" part of this forum who might be close to you. They can help walk you through your systems for free (although I might treat someone to dinner though..
6) Enjoy! Sounds like a nice trailer. Got pics?