I have a good idea of whats up here. Just to re-confirm, your nighthawk is connected, and you can access the internet from it?
I believe the problem is that your Nighthawk is handing out LAN IPs in the range of 192.168.1.X, which is also what the G1100 hands out, so the G1100 and Nighthawk network are overlapping.
You'll need to change the IP range that the G1100 uses for the set top boxes. To do this, please login to the G1100 (Easiest way to do this would be to plug an ethernet cable from the G1100 into your computer). Go to http://192.168.1.1/ and enter the username and password listed on the side of the router.
Once you've logged in follow these steps:
Click My Network at the top of the screen. Then click Network Connections on the sidebar of the page that will appear.
Click on Network (Home/Office).
Click the Settings button at the bottom of the page.
Look for the portion of the settings page which says Use the following IP address:, you should see 192.168.1.1. This is what you have to change.
Change the address to 192.168.5.1.
You just need to change two more settings before moving on. Scroll down a bit to the IP Address Distribution section. You'll see a "Start IP address" and "End IP address" section. Change these from 192.168.1.2 and 192.168.1.254 to 192.168.5.2 and 192.168.5.254.
Your configuration should look like this (images may not always show up due to them needing moderated manually):
Lastly, click Apply at the bottom of the page. The Verizon router will reboot. Reboot your set top boxes, and check for connectivity.
I'm 99% sure this should solve your issue 
Ahhhh okay.
The problem here is your router is not actually routing. Your laptop is requesting DHCP, but instead of getting a 192.168.1.x address, its getting an actual internet address from Verizon.
Please unplug the ethernet cable from Verizon going into the Netgear router. With this cable unplugged, try unplugging then pluggin your laptop's cable back into the router. Run that command again.
If you still don't have an IP address of 192.168.1.x, the router is broken. Try once more with no devices in it to hold in the reset button. Otherwise you'll have to take it back to Netgear 
Wish I could be of more assistance remotely.
Just to clarify and or muddy the waters, I'll add my network solutions over the years.
I had FiOS Internet and FiOS TV service until last summer when I dropped FiOS TV and went to using a pair of Rokus and Playstation VUE. I have never used a Verizon router (except to activate the service where it seems to be required).
I have always had a Linux firewall/router, but you could substitute that with a Linksys or Netgear router with more or less the same result. I also had a TiVo DVR with a multichannel CableCard because I didn't get along with Verizon's DVR. (It crashed a lot.) My network for around a decade looked like this:
The important thing to note above is that I had both a coax connection and an Ethernet connection from the ONT. The coax provided the video signal to the Verizon STB and the TiVo DVR. The NIM-100 was added when I got FiOS TV to serve as an Ethernet to MoCA bridge that inserted Ethernet data in a fixed range of the frequencies on the coax cable. It existed only to provide the guide information and VOD data for the Verizon STB. (The STB had no Ethernet connection at all.) The TiVo DVR got its guide information from tivo.com over its Ethernet connection. The VOD for the TiVo also came over Ethernet.
Now, with the Rokus, my network looks like this:
