So, I'm probably going to switch to Verizon Fios. They pretty much force me to use their "Gateway" combined modem/router. I've been using an Airport Extreme for many years.
What do most people do in this situation? Is streaming (iTunes on macbook pro, through router to amp) better with an Apple router?
Can I tether the Apple router to the Verizon Gateway so that I can hard-wire more devices?
I'm surprised you're not going to use your AEBS as the primary router. Why are you using FiOS device instead? I've been a Verizon FiOS customer for 12 years and using an Airport Extreme directly connecting to their ONT for about 10 of those 12 years. I'm currently looking into whether there are issues pertaining to SIP through the Gen 6 AEBS but I would not use Verizon's "modem" for anything.
DO NOT talk to Verizon sales as they don't have a clue and will absolutely sell you a router. Tell them you want to talk to technical support, preferably level 2 or 3 so you get someone who asks whether your modem is plugged into the wall or not.
When you talk to the tech, tell them you have connected the WAN port of your AEBS directly to the Ethernet port of their ONT and that you'd like them to clear the MAC address from the ONT and refresh it. Once they've reset the ONT, it will latch onto the MAC address of your AEBS and assign an IP address. From that point forward, your AEBS will be the only router you need. If necessary, you may need to add an 8-port (or more) Gig-E hub to the AEBS in order to provide IP addresses to devices like your set-top boxes for FiOS or other peripherals but from this point, consider it done.
If you want to see a network scan utility, try "iNet" from the App Store, it will scan all devices on your network and help you manage whats connected by providing visibility because AEBS doesn't.
Hope this helps...