Power dissipation is simply I2 * R so (0.3) 2 * 0.177 = 0.016 W
The IRL620 looks suitable, too. Lots of different parts will do the job.
For your diode: "This device is intended for use where low drop forward voltage is required to reduce power dissipation."
Check the reverse recovery time, that's what counts most. Main properties of Schottky diodes are the low forward voltage (largely irrelevant for your application), and the fast reverse recovery time (that's what you need) at the price of higher reverse leakage currents (totally irrelevant for you).
Normal diodes may work, too. Not the very common 1N400x series - those are quite slow. The 1N4148 is fast (1 ns reverse recovery time), but can't handle much current. Something like that one but for higher currents would work well for you.