This is a tricky question which can be answered many ways (all of them correctly). First, what is heat? If you define it as vibrations of matter, then you can create, destroy and transfer it.

Generally, however, what is meant by heat is the vibration energy (so called Gibbs free energy) and entropy.

Entropy is a measure of the randomness of a system -- for example imagine a flat pan with 50 marbles in it,vibrating a bit. If you look at the marbles, they will quickly spread out to a uniform density, independent of the starting points. Moreover,if you try to move them into a corner, they will bounce off the divider and create a force you need to do work against. This work is real and corresponds to a change in entropy of the system. In this special case, the entropy can be reduced by exerting a force and doing work.

For a more general heat system, if you include the heat used to create that force, you'll find that the entropy of the whole system (pan plus the man moving the vane) will have increased, although the entropy of the pan is reduced.

Effectively, everything is a heat engine-- so you can't escape the net increase in entropy. It is this kind of 'heat' which can only be transferred and not created or destroyed. (Although it might seem that we created entropy -- we also had the possibility of getting work for it -- so that energy conservation implies heat conservation as well.

This is a good question. One usually does not talk of heat being "destroyed" but rather, converted to other forms of energy. Heat can be converted to work. This is what happens in a car, which burns gas to produce heat that is converted to work. Not all the heat is converted to work; a lot of it is used to heat up the surroundings.