OK, so a 'cold' molecule (lower kinetic energy) coming in contact with a 'hot' molecule (higher KE) in the gas or in the container wall will start moving faster. Why?
At the cost of slowing hotter molecule. Their total kinetic energy doesn't change, it is just usually more evenly distributed after collision.
The container wall seems to be the simpler case since the molecules in the container wall are not moving (this might be an oversimplification). What about a hot container wall will suddenly make a molecule that impacts it start going faster. Why doesn't it just rebound at the same speed that it hit the wall at?
Wall is moving, vibration means motion. Wall atom/molecule can be moving outside or inside when hit by gas molecule and final result of the collision can be that some of the energy is transfered from the wall to the molecule or from the molecule to the wall. Depending on their relative temperatures energy will be transferred from the colder to hotter object. Not that transfer is statistical, for large sample, not for every individual molecule in each collision.
Throwing out the container wall altogether, what if you were to locally heat gas molecules with a laser? Let's say atom A is going in a straight line and gets heated with a laser. Does it come out of the laser at a faster speed or does it have to wait to collide with another atom before they both start going faster? Again, what specifically makes these two atoms in open space start going faster either before or after the collision? Something to do with higher energy states? It sounds like you're saying the hotter molecules vibrate more and thus result in a more forceful rebound in a collision?
Almost all types of energy we are talking about right now are kinetic. Rotation is kinetic, vibration is kinetic (although 50% of that is potential). For rotation you may treat molecules as if they were rigid. What happens when you drop someting on fast spinning wheel? It gets bounced fast at the cost of spinning wheel kinetic energy - same happens with gas/molecules. For vibration you may treat molecules as if they were made of marbles and springs - imagine what will happen to other molecule when hit by atom moving fast through its equilibrium position. Energy gets dispersed through collisions as it is statstically much more probable that energy is distributed evenly. Still, thats only statistic, so some molecules will be faster and some slower.