1. Does it matter what volume a power tube is played at? Does that effect tube life?
If the tube is operating class A1, the power its making won't matter. If operating class A2 or A3 the higher power levels will probably affect tube life. If the tubes are running class AB then they will run cooler, which could translate to longer life, but higher power will shorten that. You didn't mention the load but that affects things too- its rare that the speaker actually loads the output transformer correctly for a given tap; transformers transform impedance in both directions so a load too low on the output transformer will be a load too low on the output tube(s) as well. That will reduce tube life as more of the power made by the output section will be dissipated in the tube(s) rather than the load!
2. Does it matter if a tube is cooled, say, by a small fan nearby?
It helps! During WW2, 6L6s were used due to shortages to get amazing power levels by being water cooled.
3. If a tube is powered up but not making music does that "cost" tube life just as if you were playing music through it?
Always- how much depends on the class of operation and other variables such as dust on the envelope and so on.
4. What is harder on a power tube? Turning it on and off, let's say twice during a day (two listening sessions totaling three hours) or letting it stay on, let's say for an eight hour period?
Tubes wear out no matter what you do. They also draw power... The longer the off time the easier it is to answer a question like this. If we're talking about an indirectly heated power tube, the turn on should include time where no B+ is applied until the power tube cathode is properly warmed up. We supply a Standby switch for this purpose on our amps. If the tube is directly heated it won't matter. So if it were me I'd shut the amp down when not in use, even if for only an hour.