There's noting wrong with your basic plan, but here are some things to mention.
As much as you can, I'd try to choose assignments further north in the summer and in south during the winter. MHs, even in mild weather, can get very warm inside during the day. If your dogs are barkers, people are going to hear and be annoyed with them barking and the windows are open. Yes, you can cool the MH with the air conditioners - make sure they work and are putting out very cold air.
$15,000 for a Class A scares me a bit. Check the age of the tires and batteries. At $15K, I'm assuming it's an older gasser, probably with 19.5 tires on it, so 6 new ones will be about $3K and batteries will be about $100 each and it probably won't have more than four of them. The reason to watch the tires is the tread is probably fine, but they'll be aged out due to UV damage that you can't necessarily see. If you blow a rear tire on a MH, spending $3K for new tires would have been a lot cheaper than the damage done by the blown tire. They often ruin whatever is within three feet of the tire. This means your water tanks, electrical wiring, the MH body, storage compartments, etc.
As others said, check for roof and plumbing leaks. Check all of the closets, cupboards, roof corners, etc. Does the reefer work. Make sure it works on whatever type of power it was made for. It could be some combination of 12 volt, 120 volt AC and / or propane.
Wait on satellite internet to see if you really need it. Very few people find that it's worth the hassle or expense.
The MHs you're looking at in that price / age range probably have a max tow capacity of 3,500#. Check the tow hitch so you know. There should be a mfg's sticker on it or there may be a list of specifications inside one of the closets.
Unless the MH looks ratty (you may need to do a wash and wax job on it) or you're trying to stay in a high end resort, you shouldn't have a problem with the age issue in the RV parks. Though they could use it as an excuse to kick you out if your dogs are noisy and bothersome.
Come back an let us know how things went.
Bill
