HI,
I have a couple of thoughts:
- You cannot apply 5V to the I/O pins of a 3.3V Nano. The 5v will feed through some internal protection diodes and raise the power to the processor to 5V. This may or may not damage your Nano (not sure). I have a pro mini that runs on either 5 or 3.3 V.
- Some older garage door openers use AC voltage for the controls. Perhaps the newer one are DC. You should check.
- When connecting two systems together you must know the grounds are common. If not you should use some isolation, Relay, Opto Isolator, maybe just a Mosfet.
- I believe it is best (required) to have the drive to the garage switch remain Off when the arduino is not powered and when powering up.
Another thought, if you are automating your door in some way, it might be useful if the arduino knew if the door was open or closed. I used a simple alarm type magnetic switch, mounted at the top of the door.