1) Why do I have to wait until setup in order to access hardware? Why can't I do so when creating objects in a constructor?
The IDE creates a main() function, that calls init(), setup() (your function) and loop() (also your function).
The hardware initialization happens in the init() method. The main() function is called AFTER the constructors. Clearly, then, diddling with the hardware is useless before the init() method is called.
2) When my robot first starts, the left motor twitches 3 times-always the exact same twitch each time. Any way to get rid of this?
Set the position before you attach the servo.
3) Last but not least, what would be the best way to not need to include the Servo library in my main sketch? Seems to me that if it's included in the .h file that should be good enough, but of course this causes an error. I've looked online and there are suggestions, but I can't get any of them to work.
You can't. The IDE is designed to prevent you from hiding the use of a library from the sketch.
Well, maybe that is only a side effect of the way the IDE works, but only the cpp file that the IDE creates from the sketch is scanned for include files, and only those file, and associated source files, are copied to the build directory.