PID control of NiCr wire
Try to find some tables or other empirical data that are similar to your application. There are equations for heat loss in various mediums (King's law for heat loss from an infinitely long cylinder comes to mind), but I suspect you'll get better results from testing. You don't need to be exact- just to have more than enough power under the worst-case conditions. 
Well, you don't really have a specification until you know the current, which will depend on a number of factors, and the PWM frequency. If your PWM frequency is low then you can certainly switch the 0.5A available directly from a USB (0.1A is all that's really kosher) without a MOSFET gate driver. There is no guarantee that is enough power, however, even if 100% of it makes it to the wire, that's only 2.5W. 
You could incorporate a current measuring IC (or shunt resistor and amplifier) that is fast enough to settle during the PWM on-time and measure the resistance. You could also measure the '5V' since it's not likely exact and there may be significant voltage drop in the cable. 
Once you have the resistance you can use the temperature coefficient of NiCr and do a single point calibration (measure the resistance at room temperature without allowing self-heating.