Hi and welcome! Tires.....there is a huge amount of discussion on this, but both of our trailers have had China manufactured tires and we have had absolutely no problem with them. Check the tire pressure and torque every morning before you head out. Give them a good looking over for signs of wear. Keep your speed where you are and you will have eliminated many of the possible problems.
Re the air conditioner....I suppose it's possible that it could effect your wife's throat, but I would guess that a good, regular cleaning of the filter will fix it. You can access the filter in the ceiling unit. It's usually a piece of thin, sponge type material. I use a small amount of Dawn dishwashing liquid and water, swish it around in the sink, let sit for a bit, rinse well and let it dry. Replace. I do ours about 3 or 4 times a season, but since you are using yours daily, since it's a 5 minute job, suggest you do it every week or two.
Since you always stay at RV parks, I think an air compressor would be a waste of your money. Most of the larger Travel Centers such as Pilot, Flying J, Love's all have air compressors and even if they charge you a small amount, you're money ahead. Besides, you'd have to carry it and they are not lightweight. Definitely keep a good pressure gauge on hand for your daily tire check. Remember to check the tire pressure and condition of your spare tire also.
Agree that before you get into the colder weather areas, test your furnace. It likely will set off the smoke/CO detector, so open your doors and windows to let plenty of air in. This just lasts a few minutes and is not dangerous, but the detectors are very sensitive and will likely go off.
You certainly can travel with your refrigerator operating with the propane. It does not use very much and you will be sure your frozen foods stay frozen. It is perfectly safe and only in a few places in the country with long tunnels usually, is it required that you shut off the propane. Most of those are along the east coast.
Re your TV....if you do not want to attach it permanently with a bracket into the wall, most of the TVs have a base that lets you just set it on a countertop. The problem would be that you would not have access to the antenna if you can't set it close by. If you usually would just watch a movie, you can put it wherever is convenient.
Hope we have been able to help with your questions. It sounds like you have a great time with your travels and performing. Best of luck to you for lots of success with your talents. Safe travels.
