According to the Aerostar's specification sheet, it says "Must have
350hp engines above FL250". Is this a regulation thing, or just a
limitation of the type? I tried searching through any kind of e-FAR I
could find, but came up with squat.

http://www.aerostaraircraft.com/specs.pdf

Come to think of it, there are a bunch of altitude restrictions based
on equipment, that are spread throughout the FARs, such as the Mode C
rule above 10,000; oxygen starting at 12.5; and DME above FL240.

I also remember a King Air pilot saying in order to fly above a certain
altitude (something like in the FL3XX range) you had to have a
specially certified altimeter (which costs an arm-and-a-leg)

Is there one place that has all of these altitude restrictions listed?