First off: that is an LCD screen, not LED.
Jlawler:
I've soldered wires from the LED holes on the board to the LED holes on the LED. There no markings so I assume it doesn't matter what order I put them in
Don't do that. There is a pinout diagram on the product page that you linked. The pins have definite functions and need to be connected up correctly. If you have connect the power pins the wrong way around you've probably killed the display- best case scenario is that you haven't hooked up the power pins to the power supply of you Arduino at all, and hopefully haven't killed the display.
Jlawler:
Secondly I seem to have a shortage of buttons on my board and I would love to not have to solder more on if I don't have to. Is it possible to make robot do something like drive forward if I press button A but drive backwards if I press button A twice?
Absolutely. If I were you I would start by Googling something like "Arduino double click".
Jlawler:
Also is it possible to program reset to do something other then just reset the program? Would it even be wise to?
It is technically possible to use the reset pin as an IO, but it is quite involved and you won't be able to upload code like normal. If you have to ask then it isn't something you want to try messing around with.