Gravity is one of the four fundamental forces/interactions in the universe. It's fundamental in the sense that it cannot be explained in terms of other forces/interactions; it's an inherent property of the universe. It is equivalent to a set of base axioms in mathematics. The other three fundamental forces are electromagnetism, the weak nuclear force, and the strong nuclear force. People have hypothesized the existence of a theoretical particle called the graviton that mediates gravitational interactions, but this is more a theoretical construct to make gravitational theory a little more in line with the rest of quantum theory.
There are multiple ways to measure gravitational interaction. (I know it's cheeky, but a scale does pretty well

) Seriously, there are lots of interesting experiments. Right now, one of the hot areas of research is trying to detect gravitational "waves" from distant galaxies, similar to waves in water.
You cannot interrupt your gravity on other objects, nor can you shield yourself from gravity, based on any known interpretation of physics.
Gravity travels at the speed of light according to the theory of relativity, so if a planet far away blows up, it's fairly straightforward for you to calculate how long it will take to affect you.
Those are all good questions by the way. If you go far enough in physics, there are all kinds of wonderful things to discover related to gravity and the other fundamental interactions. If you really want to blow your mind, look into quantum theory, particularly quantum entanglement, what Einstein called "spooky action at a distance."