Can one also cool and use AC using gas as the energy source?

Sort of, one can setup a refrigeration cycle, using heat to provide the
cooling, however you still electricity to move the air.

Well, "modern" heating systems also require electric power.

(At one time many gas furnaces had "milivolt" type gas valve which used the
small amount of power generated by the pilot light on a "thermpile" to power
the thermostat circuit. If the electric was not working, the
overtemperature sensor would cut the gas. Thus, you could get a little
heat if you had gas but no electricity. Today, most gas water still work
that way so if the power fails you can still fill your hot water bottle and
take a hot bath.)


They used to have gas powered fridges for example, but they don't move
air.

Up until the 70s the gas companies were still pushing gas central air
conditioning systems. But when the gas shortages hit, they gave up on
efforts to create new demand. Without the gas companies behind them, the
makers of smaller gas cooling appliance (& AC) gave up the fight.

The "absorption" type of gas (or any heat source) water chillers (for LARGE
commercial systems) are still being made but except for large industrial
installations that use process steam they just aren't cost effective.