If you go by Voyager, nanoprobes assimilate by grasping onto blood cells and injecting stuff into them, and they control minds by directly interfacing with a humanoid nervous system. I find it highly unlikely it would work on a changeling. Or even if they could find a Founder in human form and assimilate his cells, they couldn't control his mind because their fake brain is not what does the thinking for them.
The Borg could probably annihilate the Gem'Hadar fleet but I bet the Founders would get away.
But the dominion does not really care about maintaining a empire.....
Its about protecting the founders. Now keeping a empire was the best way but if the borg turned up Im sure the founders would set the whole Gamma quadrent ablaze if needed to protect them.
That's not true. The Founders took it upon themselves to 'Bring order to the solids'. If they didn't care about maintaining an Empire they wouldn't have bothered to invade the AQ.