Multiple Airport Extremes on one gateway

What downside are you thinking about?

Think about it. If you connect the second AirPort Extreme to the first....and the first AirPort has an Ethernet problem....then the second AirPort Extreme will have a problem as well.

If you connect both AirPorts directly to the gateway (or switch, in your latest post)....and the first AirPort has a problem....then the second AirPort Extreme will continue to operate normally.

So, it would be better wiring practice to connect both AirPorts directly to the gateway. Makes troubleshooting easier as well in the event of any problems.

Gateway to 8 port gigabit switch then each Express connects to a separate port on the switch

That would be the recommended setup. Not a big deal, but you mention AirPort Extreme in one sentence and AirPort Express in another, so not sure which you are referring to.

Other ports on the switch will connect to other rooms with Apple TVs in those other rooms. Both Expresses will be in bridge mode and I’ll make all SSIDs the same with the same passwords. I’ll also ensure the Expresses are on different channels. Any downside to this setup?

Good plan. Macs will usually do a nice job of automatically switching from one AirPort to another to pick up the strongest signal from the closest AirPort as the Mac changes locations.....but....most PCs and mobile devices like an iPhone or iPad do not do this very well....if they do it at all.

If you "walk" an iPhone from the area near one AirPort to another area near another AirPort, you will likely need to get in the habit of temporarily turning off the Wi-Fi on the iPhone / iPad and then turning it back on when the iOS device is at the new location. Then, the iPhone / iPad will usually pick up the signal from the closest AirPort.

I have found that sometimes, the iPhone / iPad will switch from one AirPort to another after a few minutes...and sometimes it won't.

Can I have two Airport Extremes each in bridge mode hooked up to two LAN ports on my gateway and can they be on the same network?

Yes

Or does one Extreme need to be hooked up "under" the first?

No, but it can be if you wish. I think you already know what the downside to doing this might be though.

In their diagrams, Apple always assumes that you are using a simple modem.....not a modem/router or gateway.

So, they will always show the "main" AirPort connecting to the simple modem....and other AirPorts connecting to the main AirPort either directly or via a switch, because that is the only way that the network can be configured with a simple modem.

So unfortunately, Apple's diagrams are not very useful for most users, since most users have a modem/router or gateway provided by their Internet service provider.