Is it true in a gas kept at a constant temperature all molecules move with the same speed?

Answer 1

In general, all molecules do not vibrate at the same speed. There is a bit of complexity to this question and answer, so let's look at things a bit more closely to see what might be going on.

Different molecules have different constituent atoms and different shapes or configurations. Molecules vibrate by the compression and stretching as well as the twisting or torquing of the bonds between the atoms that make them up. There are some other aspects to molecular vibration as well, and these involve the actual make up of the individual atoms. The variety of the molecular structures we find makes for variation in the way a given molecule vibrates. As regards molecules of the same substance, we can still find some variation. Let's look more closely at one kind of molecule.

If we consider a container of water, which has H2O as its chemical formula, we'll find the the distance between the hydrogen atoms and the oxygen atom in a given molecule (which is something on the order of 96 picometers) will be varying slightly over time. We'll also find the angle of separation between the hydrogen atoms (which is a bit over 104 degrees) will vary slightly over time as well. At a fixed temperature and pressure, it may be possible to state that the average vibration of the molecules of that water are the same, but there will still be slight variations in individual molecules. This is the nature of examining large numbers of molecules of a given substance over time, even if the substance is at a fixed temperature and pressure. In cases where different molecules are combined, their vibrations can and will vary.

Answer 2

No, gas molecules move at different speeds, colliding fast molecules with slow molecules would transfer energy from the fast molecules to the slow molecules, making the slow molecule faster than it was going before.

Law of Conservation of Energy: Energy cannot be created nor destroyed, only transferred

Answer 3

Absolutely, yes. There is definitive empirical evidence. Dangerous implications regarding things like multiple sclerosis (my disconcerting precipitating reason for this knowledge) can literally kill you. Good luck.