Assistance optimizing home WIFI: AE Verizon Quantum Gateway

1. Can I use the gigabit wan port on the router to the wan port on the extreme? or do i have to use the LAN ports which limit the speed to 100mbs.

The Ethernet ports on the Time Capsule are all 1 Gbps ones. Unfortunately, the ones on the Express are only 100 Mbps. Thus the connection between the router and the Extreme should be 1 Gbps. In turn, any devices, connected to the Extreme have the potential to be on 1 Gbps.

2. What settings should the extreme be set to.

Both the Extreme and the router should be configured for a roaming network. In this type of network, each wireless router will broadcast its own Wi-Fi network, but each network will use the same network name (aka SSID) and use the same wireless security type & password. You then have the option to connect either or both Express base stations by wire or wireless to this network.

3. Can I run ethernet cable from the Verizon router to the airport express and still have the network under one umbrella in airport utility (i.e. have airplay on the speakers to the express). or do i have to run ethernet cables from the extreme to the express.

Yes, in a roaming network. You would want all base stations connected to the Verizon router in this fashion.

4. What settings will I need to change my express? (currently its on extend wifi network)

It will depend on your goal with the Express base stations. Per your excellent diagram, you may only want to use them for streaming. In this case they would just need to be configured to join the roaming wireless network.

One other thing. Before boring any holes, you may want to consider using Powerline adapters instead. These take advantage of your existing home's electrical wiring circuit to create pseudo Ethernet ones. In your case, you would need a minimum of two of these adapters. One at the location of the Verizon router, and another at the new location of the Extreme in the living room. In turn, you would connect an Ethernet stub cable between the router / Extreme and the adapter. No holes required. These adapter have come along way in performance.