Any time you apply exogenous CO2 to a liquid, you are "force carbonating". As opposed to dumping some priming sugar in, sealing the vessel tight, and letting the yeast produce the CO2.
While that calculator is quick and pretty much idiot-proof, it isn't as illustrative as tables and charts. This carbonation table makes it easier to see the relationship between beer temperature and CO2 pressure in achieving a given carbonation level.
As for time, I've never carbed a keg warm, I only carb at serving temperature. I don't think temperature affects the rate of carbonation gain if the correct pressure is used to compensate, but I could be wrong. Anyway, I'm likely more critical of carbonation quality than many, but I have the pipeline infrastructure that enables that, and it takes at least two weeks for kegs to become "good" and a third week to be "excellent".
So figure two weeks at least...