How to calculate Vce and Ic from transistor gain
How would I calculate Vce and Ic from a given DC transistor gain?

With BJTs there is no real point trying to use math with complicated formulas to uncover precise voltage levels. Reason: there are two many things that are uncertain and even simulators are lucky to get close so, all that can be realistically done is make an estimate.

Whatever voltage is at the base (set by the potentiometer and the 150 kΩ) will appear at the emitter but reduced by about 0.7 volts.
So let's say the emitter voltage (as determined by the base voltage) is 1 volts (implying 1.7 volts at the base).
That means that there will be a emitter current of 1/470 amps  = 2.13 mA.
Given that about 1% of that emitter current is base current you can pretty much say that all of the 2.13 mA flows into the collector.
2.13 mA through 2.2 kΩ (collector resistor) produces a volt drop of 4.68 volts.
Hence, the collector voltage relative to ground will be 10 volts (the supply) minus 4.68 volts  = 5.32 volts.

Vc = 5.32 volts and Ic = 2.13 mA
Of course, if you set the pot to a different position this will change but at least you have the mechanism for working this out yourself.