Gas door and trunk openings linked?

Ok, I start with a couple infobits if you don't know already:

1. The ignition key needs to be at OFF before the gas door button would work. That said, I do have a 164 that the PO (or somebody) has wired it the opposite way - the ignition key needs to be ON for the gas door button to work.
2. There is an emergency or manual pull string (red in color) inside the trunk that you can use to open the gas door. If it is not visible inside the trunk near the gas door then you need to remove the cover material on that side to find it.
3. If you manually lock the trunk with a primary key (vs valet key) and you have turned it anti-clockwise half a turn (to deny valet access to the trunk) , it will disable the electrical unlock button at the dash.

For the electrical problem, have you check fuse F3 (30A) at the G2 auxiliary fuse box?

Regarding above item (2), if that red cord has not been replaced expect it to break so have a pair of pliers handy. At least the trunk is operated via a solenoid switch.

I had the roll pin drop out of the connection between the solenoid and the catch. Then the solenoid works fine but does not withdraw the catch.

To check if the failure is electrical or a mechanical fault with the catch have someone press the release button while you listen for the relay to click. If silence then electrical, if you hear the solenoid go then mechanical.

The catch can be removed from the trunk area and the roll pin will be at the bottom of the fender if you're lucky. I re-assembled mine and it has worked fine since..

The red cord soon disappears if anyone has accessed behind the carpet in the fender area of the trunk as it is simply passed through a slit in the carpet and that is often forgotten when reinserting the trunk liner. If the roll pin has dropped out then pulling on the cord won't do anything either.