Would mold eventually just die by itself if its environment was made uninhabitable?
In the right (or wrong, depending on your point of view) environment, mold will die.

So to answer your question, the mold will die with a lack of moisture. However, there is a catch. The spores (seeds) can be preserved, apparently for more years than you and I will be alive.

Molds can come in all kinds of colors, so there is no one answer to the color question.

Also, as this is a basement, by the very nature of basements, it is unlikely you will prevent all moisture from existing or returning in that environment. Basements are typically moist, but your server farm will certainly stifle the molds ability to grow.

Nonetheless, the mold, whatever kind(s) it is, can still be a health hazard, and you are living in a home that probably pulls some air from the basement via wiring holes, furnace, etc. -- With the addition of a server farm there, you now have a continuous circulation of air.

Obviously it is up to you what you do but disturbing mold and making it airborne is more dangerous than leaving it be. Mold remediation can be costly, especially if it turns out its not a breed that poses a health hazard.

Since any mold can pose health hazards, even if just allergy based, perhaps consider getting it cleaned by a pro and take peace of mind that the server farm will stop its growth.