First off, $350 per month is pretty high even for a 2 HP pump. This could partially be due to a run time issue. 2 HP pumps tend to turover a pool pretty fast so you really don't need more than a few hours per day in pump run time. So I am not sure how long you are running your pump now, but that is the first step.
As to your other questions:
1. That depends on the condition of the wet end. If it is in fairly good condition, you do have the option of just replacing the motor which is cheaper than the full pump.
2. Going from a single speed to a two or variable speed most definitely will save in energy costs.
3. Yes and Yes. But if you want to downsize the pump at the same time and parts are not available for the old pump, then you are probably better off going with a new pump. Downsizing an old pump requires changing the impeller and sometimes the diffusor so they would need to be available.
Some additional details might help too:
1. Pipe size.
2. Any other water features; spa, waterfall, deck jets, etc.
3. Current pump run time.
4. Electrical rate ($/kwh).
5. Current control system if you have one.
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