Yes, if your contract states that you can cancel within 7 days, than there's nothing the gym can do about it. Just make sure you read the contract well and understand it all.
Also make sure you have a copy of it for yourself. If you don't, you might want to ask for a copy of it before letting them know that you're going to cancel.
You can keep following your set/rep plans (or close to it) with many bodyweight exercises. Just be creative and use single limb exercises so your body weight is harder to move.
Some examples could be single leg squats (you can also hold something it your body's still to light) lunges, but you probably don't need much more weight).
One arm pushups, chins or pull-ups if you can find something to hang from.