It completely depends on your lease agreement. There are programs that lease all the time that will have a pre-assembled contract that will probably say something like you pay $X flat rate, which may or may not cover whatever lessons you are required to have per week (I would say the majority of leases require lessons or some kind of check in with a professional), and you might be expected to pay 1/2 of regular shoeing plus vet care like spring shots.
Some people are flexible and will allow month to month, but others want to sign off a horse for 6-12 mos and just forget about it in that period. I would say people who are more flexible are leasing their personal horse. They care about it and want it back if there’s any chance of things taking a turn for the worse or it can’t be taken care of perfectly. If you have a lease that is supposed to span a certain time, there’s often a clause that if you break the lease, you’re responsible for paying the next month, too.
Full leases are a lot more risk and you’d be required to pay for a lot more. Sometimes you can work out a care lease, where you pay for the horse’s bills in full but there’s no lease fee on top of that. Usually this is in cases where the horse’s sale value is low or it’s a personal horse or schoolmaster that is looking for a break, and they’re found through word of mouth/to trusted individuals. Full leases are essentially like owning, and sometimes even more stressful because you’re afraid to break what isn’t yours. Always always have insurance, and maybe I am just neurotic, but I’m not signing any agreement that would deem me responsible for any death or injury of a horse beyond severe neglect. I’m absolutely not signing myself into any Loss of Use agreement, and I want a financial out if for some reason this horse can’t do what I want it to. If I’m paying, I’m the customer.
I’ve never heard of a one day/week lease. Why not just take a lesson? For some reason that sounds fishy to me, like maybe they want you to pay some shoeing/vet bills, too, which IMO would be ridiculous for one day/week.
I don’t see any reason why you can’t go see the horse, except maybe if there are days the owner rides and wants the horse to have the rest of the day to be a horse. I guess I’ve never been in a situation where I just wanted to go see the horse and I didn’t own it. Usually my hanging out time coincides with my rides, and I don’t have time for much else.
It looks like a solid plan up front. My only advice is to be flexible and keep your eyes peeled. Good luck!