Can a leaking heart valve be repaired and/or replaced without opening the chest?

Sometimes. Nothing available for the mitral or tricuspid valves yet, but the aortic valve can be replaced with a transcatheter inserted valve. This is not for everyone though. It is still considered rather experimental, and reserved only for those who are unable to have open heart surgery. I look forward to advances in these procedures, and hope one day there is a routine, non-ohs alternative.
Yes. There are more and more cases of so called minimally invasive surgery or endovascular surgery done today. It would depend on the condition of the patient and the type of disease being treated.