I slack off constantly on my fixed gear.

Add to that the fact that riding fixed encourages you to develop dead spots in your pedal stroke, and I'm pretty sure there is no physiological/training advantage EXCEPT for improving cadence.

Ride a lot and challenge yourself.

You don't need to do uber-long distances. You could get very fast without ever doing anything over a 40 mile ride. Eat lots and lots of hills. A nice steady diet of hills. Hill repeats, intervals. Eat lots of protein and do recovery rides in between your difficult, high intensity rides.

You don't need a power meter, though they do help once you get to a certain level. Just ride difficult terrain and push hard...you don't really need to be more precise than this until you are more advanced.

You don't need a power meter. A cheap heart rate monitor will do just fine for interval training. If you move to an indoor trainer get rollers. They will help you with your spin and bike control. Plus they are far less boring than trainers.