I do really like the skipping idea, it is a great one! The only issue I could see is that many indoor spaces have ceilings that are too low for adults to skip comfortably, while many outdoor spaces could invite gawkers

So while you are waiting to get that rope, figure out where you can comfortably do that one. For the stairs, if you have access to a building with a lot of stairs that don’t get much traffic, I’d make that how many stairs you can climb in one minute—going down stairs gets harder on the knees as you get fatter, so I wouldn’t suggest doing that in a rush.
A few more ideas (these are inspired by the Canada Fitness tests my age group had to do in grade school, but I replaced the tests with things that you could more realistically do--given that you are already fat and probably not in great shape--but which measure the same basic things)
- Wall push ups. Choose a distance from the wall to put your feet, such that with straight arms you are still leaned in toward the wall a bit. The do ‘push-ups’ against the wall. Best measure is probably how many good ones you can do without stopping, but could also see how many you can do in a minute. Just need to make sure that each time you do it the distance from feet to wall, and between your hands, is the same.
- Shuttle ‘run’ (walk, waddle….). You need a room or hallways with some length to it for this one (say at least 8-10 long steps for you). Measure a distance about six long steps for you (so that the distance will be the same each time you do this). Have three small objects (bean bags, cans of soup, whatever you can easily grab and carry in one hand), and put one of them on the floor at one end and the other two on the floor at the other end. Start behind the one object, lying on the ground in a push up position. At the start signal get up and hurry to the other end and grab one of the objects off the floor, then return it to the starting end, put it down and pick up the object that was there. Take it back to the far end, get the last object and bring it back past your start point. See how long the whole thing takes. There is getting up, accelerating, decelerating, bending down…all things that will get harder as you get fatter. Bonus for your guy if you do this in very little clothes.
- Time how long you can stand on one foot, then the other foot. Add these up, and see how this changes over time. Balancing on one foot works the core muscles, so the more out of shape you are and the more weight you have to balance, the shorter the time will get.
- (this one will leave some pencil marks on your wall). With a dull pencil held in your hand (so that it is not sticking out much) stand next to a wall, jump, and see how high you can leave a mark. Do it five times and see how high you can get. Obviously both losing fitness and getting heavier will limit how high you can jump.
- For this one you need a step or low (solid!) bench. Step up with one foot, step up with your second foot, step down with your first foot, step down with your second foot—that is one ‘step.’ Do as many of them as you can in 20 seconds (this checks your anaerobic capacity mostly, as well as strength in your legs versus weight)
- Choose some place where you can walk freely, like around a park or something, and figure out a fixed route around it. Walking as briskly as you can (even jogging, if you are up to it for short stretches) so how far you can get in fifteen minutes. This is more a test of longer-term endurance and wind.
Whatever you do, I hope you both have fun with it! (and if we are really lucky, maybe you'll tell us about it too?)