So, evidently, we can already make transistors that are a *lot* faster than silicon ones by using germanium as a dopant, or probably even just replacing silicon.. probably not economical for entire CPUs though.. but germanium has a bandgap of 0.67 eV near room temp, and silicon has a bandgap of 1.1 eV. ..but from what I'm seeing with solar cells, the more bandgap the better, and it's also colder is better, just as with CPUs.. so I know you need *a* bandgap to have a transistor, but is bigger better?

If you are trying to design electronics that require the least amount of power to process information, basically to switch the transistor on, then a lower bandgap would be better.
Congratulations to us on a fantastic breakthrough!

so I know you need *a* bandgap to have a transistor, but is bigger better?

For electronics you want a small bandgap. Germanium is a good substrate for GaAs components - but unfortunately it (and the GaAs) are rather more expensive than the current Si components.

For photovoltaics the bandgap issues are a bit more complex. The photons deliver a certain amount of energy (dependent on its wavelength), and you want to tune the bandgap to have exactly that energy (which means you need several sorts of structures if you want to capture several parts of the spctrum)

There are several major determinants of (reasonable) transistor switching speeds today. The first is the capacitance of the gate. This is determined by the properties of the gate, and the size of the active area. (Usually measured as effective gate length, EGL, which is then multiplied by the width of the transistor.) Higher voltages drive transistors to switch faster, but for a 10% increase in voltage, you have at least a 20% increase in waste heat to deal with. Recent switches to high-k metal gates would seem to be increasing capacitance, but they make it possible to have a thicker gate (which reduces capacitance) and still get the other properties you need.

The band gap compared to the operating voltage (Vcc) used to be a determinant, but with CMOS the only power consumed is in switching, not sitting there.

But the reason for using GaAs or graphene is electron mobility. Electrons have to move across the gate in switching. Graphene has an amazingly high electron mobility.