Its never a good idea to "learn new things together". If you don't know how to jump and he doesn't know how to jump, then likely you're not going to do so well. Jumping isn't just about popping over jump. Its about gaining control over stride length, learning how to collect, gain rhythm, holding a contact, and then you might look at going over obstacles. If he hasn't been taught to do these things then that's a whole lot more work. Also make sure you're riding in the correct way - jumping is different from western or dressage or trails. You'd be best to get some lessons on a horse that has been trained to jump.
As far as bonding goes - horses aren't dogs. It's rare that they get very close to people (at least in the way that we understand). To get a good working relationship with a horse I think it takes about a year of consistent (daily or so) work. Four months is a very short time.
Although when leasing horses its often good not to get too attached, as they can be taken away from you at any time. They are not your horse.
You don't seem to have a firm grasp on your plans right now so I wouldn't get a horse. The horse you need now might be very different from the one you want six months from now. Try doing some lessons doing what you want to do on horses that know what they're doing. Then decide if this is the right horse for you.
This really makes sense. I know a girl that does show jumping with her horse, so I'll ask her about her barn. Thank you so much.