What are the effects of the pull-down resistor (33K) when the solenoid-valve enters its turn-off phase?

It is in parallel with the coil, so will serve to discharge the current in the coil.

Will it prevent the solenoid-valve from ever discharging because current would flow from ground through the pull-down resistor to the solenoid-valve?

No, it will help discharge the solenoid in a more controlled manner.

Without it, the solenoid will discharge even faster, but produce a much higher reverse peak voltage when shut off.

With a lower value such as 1k, the solenoid will discharge slower, but produce a much smaller reverse peak voltage when shut off.

How do we protect the MOSFET (e.g. clamping) when the solenoid is in the turn-off phase?

If speed is not an issue, then a garden-variety 1N400x diode is used instead of a resistor. This makes the coil open very slowly, but blocks the peak reverse spike to -0.7V or so.

If Ids Max is 2A and the solenoid IPeak was 25mA (derived from 0.35W solenoid spec), should I even be concerned about protecting the MOSFET during the turn-off phase?

Absolutely. The inductance is stated as 32 Henries, which is a lot of inductance. Such an inductor could make spikes in the tens of kilovolts range. Without some means to clamp that spike, it will destroy the MOSFET.

What are the effects of the pull-down resistor (33K) when the solenoid-valve enters its turn-off phase?

It is in parallel with the coil, so will serve to discharge the current in the coil.

Will it prevent the solenoid-valve from ever discharging because current would flow from ground through the pull-down resistor to the solenoid-valve?

No, it will help discharge the solenoid in a more controlled manner.

Without it, the solenoid will discharge even faster, but produce a much higher reverse peak voltage when shut off.

With a lower value such as 1k, the solenoid will discharge slower, but produce a much smaller reverse peak voltage when shut off.

How do we protect the MOSFET (e.g. clamping) when the solenoid is in the turn-off phase?

If speed is not an issue, then a garden-variety 1N400x diode is used instead of a resistor. This makes the coil open very slowly, but blocks the peak reverse spike to -0.7V or so.

If Ids Max is 2A and the solenoid IPeak was 25mA (derived from 0.35W solenoid spec), should I even be concerned about protecting the MOSFET during the turn-off phase?

Absolutely. The inductance is stated as 3.2 Henries, which is a fair amount of inductance. Such an inductor could make spikes in the kilovolts range. Without some means to clamp that spike, it will destroy the MOSFET.