Is there such a thing as Negative Absolute Zero?

Yes and no. You are both right but for the wrong reasons.

In recent discoveries, they have in fact measured negative Kelvin!

But negative Kelvin is just positive Kelvin - that is, a measure of atomic kinetic energy, which is always positive. The difference is that in positive temperatures atomic motion is random, whereas in negative temperature atomic motion is uniform - they create a system of quantum locked particles, which remain uniform and orderly, and become more-so as energy is added.

Generally speaking though, you cannot get negative Kelvins.

https://www.youtube.com/watch?v=APFqPoXBE2o

Sure. It is possible in systems with limited states, for which the entropy decreases as energy is added to the system (at least, that what I can get from the first reference).

This is a consequence of using "thermodynamic beta" or "perk" instead of temperature to describe a system. Perk is more fundamental than temperature, and is related to the reciprocal of temperature. (Third reference.)

Note that negative temperatures are "high". The next temperature higher than +∞ K, is -∞ K, and the temperature scale continues up to the very highest possible temperature, -0 K.

It all sounds very strange to me, but statistical mechanics was never my strong suit.

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The second reference is less technical and seems a bit easier to follow.

By definition, zero Kelvin is the lowest possible temperature.

The Kelvin scale is called absolute for two reasons. One is Kelvin's, that its formal character is independent of the properties of particular materials. The other reason is that its zero is in a sense absolute, in that it indicates absence of microscopic classical motion of the constituent particles of matter, so that they have a limiting specific heat of zero for zero temperature, according to the third law of thermodynamics.

It is not possible to have a temperature that is lower.