1/2 right only.
Intercooler is not merely a devise. It's something like a
radiator.
For a turbo intercooler: When the exhaust propels the impeller to
turn, it in turn rotates the turbine connected to it. When the
turbine rotates, the air is sucked in faster thus improving the
introduction of fresh air and expelling of exhaust fumes. When the
air is sucked in, the air is too hot and thus hinders good
combustion. Thus the hot air runs across a circulation of coolant
in pipings/fins to reduce the temperature of the air. This helps to
give better combustion even though with force fed air.
There're cheaper ways to make force fed air. SRAD (Suzuki Ram Air
Direct) a very old way of using the air gushing through the air
pipe when the motorcycle runs. The gushing air is forced through an
air filter before feeding into the throttle body & into the
engine. In this way, the air pipes facing the front whereby it
collects the gushing air when the motorcycle is moving becomes a
form of turbo. But because there's no hot air, the intercooler is
not needed.
For a gearbox intercooler, coolant is circulated inside
pipings/fins while gearbox ATF is outside the pipings/fins. In this
way, the coolant which has an advantage of going to the radiator to
dissipate heat collects the ATF heat and dissipates it at the
radiator.