Ever use a pump to inflate a tyre? I've given myself small burns by underestimating how much heat was released by gas compression. When you release the pressure, the gas cools, like when using a spray can - it can also get very cold.

Different gasses will change temperatures at different rates when pressurised, this is given by the Joule Thompson coefficient Hydrogen is a notable one, as it has a negative coefficient (i.e. it does the opposite of most gasses) near room conditions.

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Sodium acetate is the chemical in the reusable handwarmers.

You can coat a reactive chemical in a protective coating, then like popping a balloon, the pressure breaks the coating and triggers the reaction. The easiest example of this is putting an oxidizer and a reductant in the same bag.

There are lots of exothermic crystallization reactions where the pressure nucleates a crystal in supersaturated solution. You can control that pressure using engineering of the container, but it's not unique to the actual chemical.

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For all things not including (Hydrogen, Neon, and helium) they usually will not cool down when compressed. That's not how pressure and temperature are related for most substances.

A perfect example is the ideal gas law PV=nRT

Where:

P= pressure of the observed system

T= temperature of observed system

As you can see, Pressure becomes divided by Temperature or vice versa (depending on what you're solving for), which means they are directly proportional (instead of inversely proportional, where if pressure went up, the temperature would drop)

Now, I can go more into detail as to WHY this physically happens (hint, temperature is a measure of energy in a system, as you compress a system things, they bump more, create more friction, etc.) but I did not want to just unload a ton of info if you didn't need it to answer what you are actually asking

Edit: accounted for the Joule Thompson effect as described below in comments