That's a lot of questions with big answers. :-) Maintenance... I think you will find that most places that will do work on your coach are pretty used to accommodating Rv'rs on the road. "Most" have no problem with you spending the night in your rig on their lot and doing work during the day. "Most" will also generally run you an extension cord for basic electric and a water hose. If you elect to hotel it, I would imagine they wouldn't have any problem giving you a ride to your hotel and back again when repairs are completed.
Mail... I use a mail forwarding service. There are a number of options there. They sort my mail for me based on my previous instructions, notify me electronically when mail is received.. including the senders information. From there I have the option for them to scan and send it electronically, call in and have it read to me, or sent to my current location via USPS. I really don't receive much paper mail and none of it is very pressing (year end stuff), so I generally let it accumulate and have it all sent at one time.
One thing to consider when selecting a mail forwarding service is to see if they are able/willing to help gather and submit various paperwork on your behalf when registration renewal/tax/etc times come around.
Residency... I chose residency based on the usual. Taxes, insurance, licensing and registration ease and cost, and the fact that I had a family member living in the state. It made it easier to make the trip over to establish residency when you can pay a visit to family at the same time. :-)
Some things to consider when establishing residency with a mail forwarding service.. not all States will allow using a mail service for such things as banking or receiving benefits. It's important to do your homework.
That's all pretty much in a nutshell, but there are tons of resources online when planning for full-timing. I would recommend browsing through the Escapees RV Club
forums. Many of their membership are retired full-timers/snowbird/sunbird folks. They regularly discuss just such issues and would probably provide you with the most concise information within a single source. You don't have to be a member to join their forum.
Good luck and happy travels.