Controlling an RC Car's remote with a Mega 2560 board

If I understand correctly, you're proposing a transistor between bottom forward signal contact and ground, and control the transistor with the arduino?

Exactly or you can use mini relays, either way will work. Give me a sec to post a picture.

Added: no matter how much I crop and resize the pictures, they always end up huge.

This is an example of what you should do. (The cathode represents the forward pin on the chip)