I wouldn't pull up all the flagstone. I would continue what you are doing by pulling up the affected stones and backfilling with sand.
I'm more worried about the termites. I think you are right to be concerned about them invading your house so you should find out right away if you have them. You can have a company come out and tell you if you have termites, but a lot of companies will tell you have termites whether you do or not. They are looking to have you pay them to eliminate the termites. Either find a highly reputable company or buy some termite traps. If used correctly the traps will let you know if you have termites.
To keep your pets from getting into poison, you could put termite bait in the empty tree ring and then cover it with a piece of plywood for a couple of months or however long it takes. And everytime you pull up a piece of flagstone, dig down to the hollow area and put some bait down there.
You don't want to use a spray, use a dry bait that the workers will take to the the queen.
Same for the rats, put down poison and cover it up.
