I have a Logitech gamepad that is recognized as a joystick within Dark Forces. Is there any way for me to customze the controls?

If you run SETUP.EXE in DOSBox you can remap the joystick/gamepad controls.
Alternatively DOSBox has its own mapper to remap the keyboard or joystick: you can launch the mapper by pressing CTRL+F1 while DOSBox is running (i.e. while the game is running).

Another solution is to use an utility that enables your gamepad to emulate keyboard and mouse input: AntiMicro, JoyToKey or Xpadder.

Both the setup menu and the dosbox control editor don't like to recognize the gamepad's right analog stick. I was finally able to get the dosbox editior to map to the up and down of the right analog, but never the x axis. I was able to map the left stick to go for forward backward left and right, but moving forward causes Kyle to move at double speed, and makes an earsplitting sound of the music playing at super speed.

Are you using the Logitech F310? If so, try moving the switch on the back to change between XInput/ DirectInput. Directinput usually works a bit better with older dos games, as the Xinput standard didn't come about until the xbox or xbox 360.