I'll try to answer just a few of your questions.
1. I'll also suggest panniers instead of a trailer. There's nothing wrong with a trailer but I prefer hauling everything on the bike. Take what you need, but no more. If you pack too heavy, there's no disgrace in mailing excess stuff home when you're on the road.
2. Use highway maps to plan your route. You'll be able to determine which roads are busiest and which are less traveled. When you're traveling through mountains, you need to know how high your passes will be. Topographical maps will give you a good idea of how a road elevation will change and what you can expect to find along the way. For camping stops, most towns of any significant size will have private or town-run campgrounds. If you stop at visitor information centres along the way, you can inquire about camping facilities in the vicinity. You can also find out a lot of information about road conditions and elevations.
4. I don't use bike tights. I prefer touring shorts. They are comfortable for riding but they are also loose enough to be appropriate to wear in conservative communites where bike tights might be out of place.
J & G Cyclewear makes good quality touring shorts. For colder weather, you can wear sweat pants over your shorts.
8. Your biggest weather challenge will be when you reach the mountains. Once you've roughed out your route, make some calls or send some e-mails to state tourism agencies to ask about the weather when you're going to pass through. The mountains can get a lot of snow and cold weather. Be prepared. There are areas in Colorado where snow and sub-freezing temperatures are possible more than 300 days a year. Parts of New Mexico, especially in the mountains, can also get rough weather.
9. Your trek is quite a long one and you're traveling through a number of climate zones. You may wish to start with the gear that is appropriate for the first part of the journey. Then, when you move into a drier and higher area, have some cold weather clothing such as a winter weight jacket, mittens and a toque mailed out to you.