"So, under your specified conditions, the IR energy would be lost and therefore the kinetic energy also would be lost, then in theory eventually the molecules would have no kinetic energy."

Yes, but a clearer way of saying this is that the molecules would eventually reach absolute zero [0 Kelvin or −273.15 C].

There are a number practical problems with trying to do this- "by putting the sphere" in cold conditions and low pressure it's possible that some of these technical problems could be diminished somewhat.

Conversely, one could theoretically put "this glass sphere" in a furnace and sea level pressures, and attempt the same thing.
Or similarly- and this generally points to how one refrigerates anything- you pressurize gas, the heat is dissipated and one gets colder gas. But radiating heat via electromagnetic radiation [infrared] opposed to conducting/evaporating/or forcing air or water thru a radiator, is an inefficient [slow and "harder"] way to lose heat.

One technical problem with trying to reach absolute zero is that high speed particles could penetrate thru a meter of material and a there is the general a background of radiation anywhere on earth. And at higher elevation one has more high speed particles compared to a mile underground. [The high speed particles are going to cause more of cooling problem as one tries to get near absolute zero as compared to relatively warm condition of say -100 C.]

"Now consider the interactions between the molecules if some of the molecules were radioactive, what would happen to the Alpha. Beta, & Gamma rays?
And what would happen to the kinetic energy of the molecules?"

Trying to make any radioactive material reach absolute zero would be a problem. And Probably if someone is trying reach absolute zero, they do so in environments which have a low radiation background- a basement might do.

Also some other things:
Neutron:
"For other types of radiation, e.g. alpha particles, beta particles, or gamma rays, material of a high atomic number and with high density make for good shielding; frequently lead is used. However, this approach will not work with neutrons, since the absorption of neutrons does not increase straightforwardly with atomic number, as it does with alpha, beta, and gamma radiation. Instead one needs to look at the particular interactions neutrons have with matter (see the section on detection above). For example, hydrogen rich materials are often used to shield against neutrons, since ordinary hydrogen both scatters and slows neutrons. This often means that simple concrete blocks or even paraffin-loaded plastic blocks afford better protection from neutrons than do far more dense materials."
http://en.wikipedia.org/wiki/Neutron

And from same reference:
"Outside the nucleus, free neutrons are unstable and have a mean lifetime of 885.7±0.8 seconds (about 15 minutes), decaying by emission of a negative electron and antineutrino to become a proton"