A number of places have already declared that they allow camping. On ACA routes, the maps will identify these places. If you go to one of them, you can certainly camp there. That would always be my first choice. Some are free. Some charge.
Using the AC maps for a while to pick places to stay is a good way to get a feel for what works and what doesn't.
I have set up in city parks without asking. It usually works fine.
Yeah they are some of my favorite places to camp. It works best well away from either coast and in smaller towns.
This is a situation-by-situation thing, and you have to use your best judgement and experience. Watch out for hazards: sprinklers, dogs, bulls, falling trees, floods, etc. In all situations, leave no trace.
In much of the west anywhere green probably has sprinklers that come on in the middle of the night.
There are many articles on panniers vs. trailers on the web. A Google search will yield articles that address pretty-much all the pros and cons there are. A good measure of personal preference is also involved. For touring bikes, I think panniers work best. For non-touring bikes, a trailer would often be a better option.
My advice would be to pack light and skip the trailer. You really need surprisingly little to camp and cook.