1. If temperature is a measurement of average Kinetic Energy, what is the temperature of vacuum?
One doesn't determine the temperature of a vacuum. Just as 'nothingness' has no color, taste, smell, etc. it also has no temperature. That is because, as you point out in your question, there are no particles whose kinetic energy can be measured or averaged.
Only objects within a vacuum can have a temperature, and that temperature will depend on the balance of incoming and outgoing radiation. Electromagnetic radiation can travel through a vacuum, so objects in space of any temperature above the near absolute zero (0 Kelvin = about -273 deg C.) temperature of cosmic background radiation (which is about 3 Kelvin) will radiate energy into space. Without another source of energy replacing that loss (a nearby Sun, for example) the object's temperature will decrease. That is why you read about 'the coldness of outer space'.
2. Would my hand feel hot or cold in vacuum?
Your hand 'feels' hot or cold when heat is transferred to or from it, respectively. There are only 3 ways that heat can be transferred--by conduction, convection, or radiation. Only the third (radiation) can take place in a vacuum. That is how the Sun's energy is able to reach the Earth.
What your hand feels in a vacuum depends on the radiation falling upon it. For extreme examples, the vacuum a few thousand miles from the Sun's surface would feel very warm as it absorbs highly energetic photons. The vacuum in interstellar space, however, contains only background radiation close to absolute zero (-273 degrees C.). In that situation, your hand at body temperature loses more energy than it gains, and would feel cold.
Any material object at a temperature above absolute zero radiates energy. In the case of a vacuum chamber, the temperature of the chamber walls will determine the radiation environment. If they are above the temperature of your hand, your hand will absorb the radiation and feel warmth. If they are below your hand's temperature, your hand will feel coldness.
3. If you had an iceberg in a bathtub and it melted, what would happen to the water level in the bathtub, and why?
An iceberg in a bathtub? It depends on whether you are considering a regular sized iceberg or a VERY small one in a normal bathtub. That's because it depends on whether the iceberg is floating or resting on the bottom. If the bathtub is large and deep enough so that the iceberg is floating, the water level won't change.
A floating object displaces an amount of water equal to its own weight. Since water expands when it freezes, one ounce of frozen water has a larger volume than one ounce of liquid water. A completely submerged ice cube weighing one ounce, for example, displaces MORE than one ounce of liquid water. The cube will rise until the volume remaining under the surface displaces only one ounce of water.
If you could remove the ice cube and leave a 'hole' in the water where the cube used to float without disturbing the surrounding water, that hole would take exactly one ounce of liquid water to fill. Let the ice cube melt. Since it is now one ounce of liquid water, putting it back into the 'hole' will exactly fill it and leave the remaining water undisturbed.
If the iceberg is sitting on the bottom of the bathtub and NOT floating, then the water level will rise as the ice melts.
4. How can you boil a liquid without heating it? Why is this possible?
The boiling point of a liquid depends on both temperature and pressure. As pressure increases, so does the boiling temperature. Pressure cookers are used in cooking to raise the temperature at which liquids within will boil. Conversely, the lower atmospheric pressure on a mountain top makes it harder to get boiling water hot enough for good tea or coffee.
Boiling occurs when a liquid's molecules have enough energy to break free from surrounding molecules. Think of higher pressure as making that escape more difficult by offering a counteracting force.
Water boils under normal atmospheric pressure at 212°F (100°C). Imagine it at 221°F (105°C) but NOT boiling in a pressurized container. If the pressure is quickly reduced, the 221°F (105°C) water at normal pressure will now boil.
For another example, put water at room temperature into a vacuum chamber and begin removing the air. Eventually, the boiling temperature will fall below the water temperature and boiling will begin without heating.
5. What would you consider to be the 5 or 10 most important/fundamental questions physicists are trying to answer currently?
That's a subjective question, but here is my vote for the most important questions science currently has before it: