Only destruction of the brain kills zombies. So all you need is one well placed shot. I would highly recommend a shotgun, because even though the chance of skull penetration is low, it acts as a solid wall of bullets, effectively slowing down multiple zack.
That depends on the origin of the zombie. Modern zombies are almost always biologic (created by a virus or bacterium). They are alive, just lacking any higher brain function, and incredibly cannibalistic. Movement in the human body requires energy derived from metabolizing food. For nutrients to be carried from the digestive tract to the muscles, the heart must also be working. Therefore, enough damage to the heart will still kill a zombie. And while a zombie can store enough ATP in its muscles to let it starve for a little while, oxygen is required on demand for movement. Damaging the lungs prevents the muscles from getting enough oxygen to perform any function. So a frangible round to the center of mass will stop a modern zombie.
If they are supernatural in origin, however, we are all screwed. Supernatural creatures don't even need a head (ala Sleepy Hollow) so good luck figuring out what to do. In that case, I'd be going to Rome and trying to convince the Pope to check the secret archives for anything that can kill supernatural creatures. And who knows what the Vatican is hiding? They don't let anyone look, ever. So they may have something that could help.