Can I power a MOSFET from my uC in this way?
You're (worse than- because of voltage drop in the output) halving the gate voltage, the particular MOSFET you mention is only guaranteed as low as 4.5V Vgs, so you're going to get more (perhaps much more, and relatively unpredictably) Rds(on) and thus I^2R heating during conduction than if you used the full 5V drive. 

Secondly, you need to consider the gate charge to determine the switching time with a load, not just input capacitance because of Miller effect- your numbers will be optimistic by a large factor because the gate-drain capacitance couples the drain voltage swing back to the gate at a multiple because the drain voltage changes more than the gate voltage. From the link above: 



There are many papers on predicting switching times, or just get a SPICE model and look at "typical" results to get a feel for it (keeping in mind that there will be a spread in parameters  in reality- not all devices are "typical"). 

The MCU naturally drives the output push-pull- you're better off driving it that way with the full 5V (series resistor) and adding a driver circuit if the switching is not fast enough to meet your requirements. A (very) rough simulation for the MCU output with a 5V supply is a voltage source in series with a ~100 ohm resistor (maybe 200 ohms for sourcing current vs. 100 for sinking). If it turns out that guesstimate resistor value is fairly important, your circuit may be too dependent on characteristics that are not guaranteed (in reality the outputs act sort of like a constant current source over a range, and that current is not well controlled). 

Here's an old Microchip output curve, the part you are using will likely be different, but the shape will be similar: 



Edit: Suggested simple gate driver. You can also buy gate driver chips with much higher gate drive, and several manufacturers have many products (including Microchip), but for 2kHz it should not be necessary. You can optionally put a 1K resistor from MCU output to gate to get the tail of the 0/5V. 







simulate this circuit &ndash; Schematic created using CircuitLab