Does Hot stuff get Colder than Warm stuff when cooling?

No. It’s entropy, man.

Everything should even out at whatever the room temperature is, I’d think.

It’s contrast, and maybe a slight effect of the container (i.e. mug vs. glass). All substances, given enough time, will equilibrate. In this case, the liquids will all eventually reach room temp, regardless of starting temp. In order for it to get colder, it would have to reach room temp, and then proceed, at that point, to heat up something else as it continues to cool. However, if everything around it is room temp, it has no where to dump the energy, and so stays at RT.

It could become cooler, as it’s entirely possible for two molecules at the same temp to spontaneously transfer energy. However, for this to happen over the scope of a whole drink, and to an extent one can tell, would be exceedingly rare. A “Thermodynamic Miracle”, as Dr. Manhattan put so well. Essentially, it simply doesn’t happen. Plus, a cup of tea would be no more likely to undergo this then a glass of tap water, say.

Also, the coffee may have some property that makes it seem colder. Things can seem colder and hotter to our senses because of various things like conductance and specific heat. Take stone, which usually feels cold at RT, though it is likewise at RT, while plastics tend to feel warmer.

Lastly, the fact that the hot water freezes fast is a completely unrelated phenomena dealing with entropy, as @gailcalled said.

For more cold water vs hot water fun you can read this

I guessed it is a contrast thing since it doesn’t seem plausible that cold could transfer heat to a warmer object, great answer though! Interesting about the conductance of particularly hot beverages.

It’s called the Mpemba Effect