Very hard for me to make out the details especially on the first two pics - blurry.
Are the blue cables ethernet cables going to room wall jacks? The yellow cable appears to be an ethernet cable (568B) - not sure. In any case, the wiring panel is well below acceptable standards and should be redone. Go online and google words such as "ethernet punchdowns", "how to install ethernet cable", "ethernet wall ports", etc.. You will have continual problems once connectivity is established (if at all possible) with the current wiring conditions. Did the Verizon techs pull and punchdown the ethernet cables?
Is this your modem/router?
http://www.ebay.com/itm/like/262575129001?lpid=82&chn=ps&ul_noapp=true
Here is a link regarding the ONT connection. However, I am not sure about the options you may have per se.
However, either way a connection (coax or ethernet) from the ONT to the modem/router is needed. Via the coax connection or the white WAN port if I am following it all correctly.
You would plug in the two desktops and the Google Wifi using three of the yellow ports on the modem/router.
All of the devices on your network will look to the modem/router to obtain a DHCP IP address. Whether that modem/router is yours or Verizon's.
For now, I suggest that you stay with Verizon's modem/router until you really have everything up and fully running. Otherwise any problems will be blamed on your device(s) and/or infrastructure.
First get your desktop PC's up and running via the g1100. Then focus on adding in the Google Wifi. The overall concern is getting the IP addresses all as they should be. Especially if you want the Google Wifi to be a separate network. Not sure about the defaults therein but I will look for the user guides and manuals.
See if you can put together a simple sketched diagram of the wiring to and from the panel shown in the photographs. That will help keep everything clear. May take some effort to get it all "untangled".