Definitely find out what all is included in the lease price. Are you responsible for regular farrier & vet visits? Are you responsible for emergency care if something happens to the horse while you are working with it? What's the horses' agreed upon value should, heaven forbid, it die while you are working with it? If you have an issue (he's lame, he's bleeding, or you find a jagged tooth interferring with the bit, he threw a shoe) what's the protocol - do you call the owner, the vet, the barn manager?
When, where & how much are you allowed to ride? Are you allowed to show, jump, trail ride, trick train, trailer, feed, clip, bathe, blanket, use spurs, use snaffle bit, use curb bit, use splint boots, tie to a picket line, tie in cross ties, ride bareback, run barrels? Is the schedule of availability or any of the other specifics negotiable?
Is anyone else co-leasing the same horse or will they be letting their cousin's neighbor's nephew's best friend ride it when he visits on his birthday? Are you using your equipment or theirs? Where does their/your tack go? How is it cared for, what happens if their saddle goes missing and you were the last known to use it?
I recently finished a lease where I had pretty much free rein over what I could do with the horse. The owner and I texted or spoke regularly and I think communication is the major key to a successful lease. Have fun!