Correct setup for my personal network

All of the AirPorts will be in Bridge Mode, so it is likely that they will all appear side by side horizontally when you open AirPort Utility.

The reason for this is that each AirPort is receiving their Internet connection from the gateway......not from another AirPort......even though one AirPort might be connected to another. So, you won't have a "main" AirPort in this type of set up, they are all equal on the network.

If you want to have an AirPort set up as your "main" DHCP and NAT router, then you will need a simple modem........the kind with only one Ethernet port.......not a modem/router or gateway. Something like this:

Your Internet Service Provider (ISP) may or may not offer or allow this type of modem with their service.

I have an AirPort Extreme 5th gen connected to my modem (internet)

It would be much better......performance wise....... if the A1521 were your "main" AirPort on the network, since it is the most up to date AirPort that you have. Is that a possibility?

We also need to know the make and model number of the device that you call your "modem", since it might actually be a modem/router or gateway device. If this is the case, then the DHCP and NAT setting on the "main" AirPort would NOT really be the correct setting to use with that type of device.

The fact that you are seeing a Double NAT error when the AirPort is set up using DHCP and NAT is a strong indication that your "modem" is really a modem/router, also known as a gateway. Double NAT is often not a serious error on a simple home network, but unless there is a special reason why you want to use this setting, it does make things more complicated. Normally, you would want to avoid a Double NAT error on a network if possible.

I then have a AirPort Extreme A1521 connected to my 5th gen.

Will the AirPort Extreme connect to your "main" AirPort using a wired Ethernet cable or a wireless connection? The wired connection is far superior to a wireless connection since you will lose at least half of the potential speed on the network with a wireless connection between the AirPorts.

.Finally I have an airport express connected to my A1521.

Same question. Will the AirPort Express connect to the AirPort Extreme using a permanent wired Ethernet cable connection or will it connect using a wireless connection?

Once we understand what type of modem that you have, and whether you are able to make the A1521 AirPort your "main" AirPort, and how the other AirPorts will connect to the "main" AirPort, we'll be able to suggest the correct setup.

Please keep in mind that the correct setup may require you to reset each AirPort and set it up again, since changing settings on the fly is not the most reliable way to go.

Sorry, although the picture you provided does indicate that this is not a simple modem, it doesn't tell us the actual make & model. It may be printed on one of the sides or on the bottom of the unit.

I’m sure the modem etc is a gateway:

It appears to be, so you would set up each of the AirPorts in Bridge Mode, which Apple calls "Off (Bridge Mode)". With this type of setup.....assuming each AirPort connects using Ethernet.....it would make sense to place the A1521 AirPort in the location where most of the WiFi devices will be used most of the time.

Locate the AirPort Express where the least amount of WiFi devices will be using it for their connection.......since the AirPort Express is much slower than the other AirPorts.