Answer 1:
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.

Answer 2:
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.
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