Heat and space

A mass of gas has its molecules moving. (Say in a jar)

Another mass of gas has its molecules moving faster. This gas is hotter than the first one.

So, heat is motion of molecules.

Now, I take this mass of gas into deep space devoid of any gravitation.

My questions are,

1. Will this gas cool down. If yes, why?

Let me give my detailed insight/conjecture.

A body moving in uniform motion does not get affected by the extremely low temperature of deep space and continues its 'motion' unabated.

Each molecule is in motion, even if not uniform motion.
So, if the gas cools down, why does deep space affect the motion of molecules?

2. Does deep space know the difference between uniform and non-uniform motion?