1. At absolute zero, things have fallen into the lowest energy state that they have. Infinite temperature would mean having an equal chance to be in any of the possible states. However, the states just keep going on up to higher and higher energy. For example, you can have states with any number of photons. As the chance of being in any state gets close to equal, the average energy blows up toward infinity. So with some finite amount of energy around, the temperature stays finite. With more energy, it could always get larger, so there's no upper limit.
2. A vacuum can have any temperature. The temperature determines how much random thermal radiation, mostly electromagnetic waves, is present. The vacuum of distant space happens to have a temperature just under 3K, very cold but not as cold as one can reach in an ordinary lab. At temperatures over about a billion K not only electromagnetic waves but even electrons and positrons start to show up. I guess at that point, you might want to stop calling a vacuum by that name.
3. Assuming the ring were made of similar material, it would also expand, making it easier for the ball to pass through.
4. Usually the heat is flowing out to the cold air from the surface. The bottom is usually warmer than the air, if it's been sitting under liquid water. It helps that when water gets below 4°C, it actually expands a little, getting less dense. That means that the coldest water can just float on top, where it can cool down even more.
(published on 10/22/2007)