How does an army of slave soldiers work? Are the leaders free? While the Unsullied are special with their conditioning, what’s to stop the Tiger Cloaks from seizing power?

Yeah, the officers are free:

“"I know of Valyrian steel,“ said Dany. "Ask the Good Master if the Unsullied have their own officers.”
“You must set your own officers over them. We train them to obey, not to think. If it is wits she wants, let her buy scribes.”

As for the Tiger Cloaks, I would imagine it comes down to the officer corps being freeborn, a carefully-inculcated internal culture of pride and discipline, and better treatment than other slaves.