This is to an extent a party trick to befuddle lay people. Physicists know perfectly well that temperature is not a well-defined concept out of equilibrium. And when in a population inversion experiment when "temperature" is determined to be negative (or "beyond infinite" if you will) it arises because for a short while you have a non-Boltzmannian distribution.
Right, using beta is more convenient and also it's better behaved since it doesn't have the weird "goes to infinity then comes back from negative infinity as you keep increasing it" behaviour.
Oh, right, yes. But that's just because 1/T occurs in many formulas and thus it's often convenient to work with it instead. No exciting new physics hiding there. ;)