Thanks to you both for replying.
To @outsider I had my multi-meter set on DC when I got the 7.7v. I had checked it first for AC but don't remember the number that displayed except that it was somewhat close to this same number. Maybe 9v (ish) but can't say for sure. I figured if it was AC, and was that low, checking for DC probably wouldn't kill my meter. 
I've attached the owners manual. It's a Genie Pro CM8600IC (1/2 HP motor) which is usually simply referred to as a CM8600. See page 33 and the switches I'm referring to are connected to terminals 15 and 16 on that page. It looks to me that it's getting power from the secondary winding of that coil which is on the neutral side of the 120v AC circuit. I can't remember, does power on the secondary side of a coil always end up as DC?
If this is DC, I see from the diagram that the RPM sensor is also powered from this same secondary side of the coil and I was already hoping to tap into that RPM sensor (rotary encoder) for a signal to display the percentage that the door is open. I know that's somewhat redundant with what I'm trying to do with the switches (above), but I'm expecting the encoder to not always be spun by the exact same count every time. Maybe it will, but if the sensor is already there, why not use it is my approach. That way, I'll have backup information about the state of the door. Do either of you (@outsider or @wvmarle) see any issues now that you can see the wiring diagram for my GDO that I might have trying to set that up? I was thinking just tap into those and follow other efforts in these forums for connecting an encoder to an Arduino.
To @wvmarle I've heard of optocoupler's, but that's about it. Can you explain what they do and how to use them here? Are they similar to diode's? I have a mechanical engineering degree but that's a far cry from an electrical one. I'm definitely out of my element. 
But I've always enjoyed dabbling in electric stuff too. This is my first "smart" project though.
Again, thanks to you both!
Genie Pro Screwdrive (full manual).pdf (1.16 MB)