Electrical – Power Dissipation VS Switching Power

A single transistor will not provide an interface with your FET AND do it without inversion. You can do the job with two transistors, though.

^{simulate this circuit – Schematic created using CircuitLab}

This will provide a decent drive for your FET with about a 1 usec added to on-time. That is, if you put in a 10 usec pulse, the FET will be on for about 11 usec.

Also note that R5 needs to be a 1-watt resistor.

EDIT -

It turns out I was using the wrong FET model, and my value for gate capacitance was too low. For a "real" IRFZ44, the added pulse width is about 3 usec, not 1. The problem is that, when you go to turn off the FET, The charge stored in the gate can only discharge through R5, and the time constant is in the 3 usec range. You can, of course, decrease R5, but then the power it dissipates when the load is driven on goes up. You can get a 2 usec delay rather than 3 by decreasing R5 to 100 ohms, but then you need a 2-watt resistor. If you need very narrow PWM cycles (low effective motor current) you're probably better off with a more sophisticated driver. I personally tend to go with the Maxim MAX4426/4427 (less than $4 each, with 2 channels per IC), but that's just habit.

You have mostly thought this out correctly.

One problem I see in your figures is the power dissipation of the FET. You say the gate will be driven by a 3.3 V digital output. The only guaranteed on Rdson is 135 mΩ at 2.5 V gate drive. You can't use the 105 mΩ figure since that's for 4.5 V gate drive.

The power dissipation will be (500 mA)²(135 mΩ) = 34 mW. You will have difficulty noticing that get warm. It's not anywhere near any limits.

Inrush current will be much larger than the steady 500 mA operating current. However, it won't last long, so it looks like you'll be OK. Note that the maximum pulsed current this FET can handle is 6 A. That sounds sufficient at 12x the operating current. Of course the dissipation will be quite large then, but again, short lived. Without details on the motor, we can's say for sure, but I think you'll probably be OK.

It's good that you put R9 there, but I'd make is smaller. 100 kΩ will do a better job against noise that might be picked up, and is still of no consequence to your digital output.

If you plan to switch the pump on and off rapidly, I'd use a Schottky diode to get around reverse recovery time issues. If you always leave the pump off for more than a millisecond or so, then this doesn't matter.

I would also put a small cap across the pump, especially if you aren't going to switch it rapidly. That will help with RF emissions.