Having the EE router issue IP addresses with all the AirPorts in Off (Bridge Mode) is the simplest way to set up the network. This is probably OK....assuming that the EE router can provide enough IP addresses for all of the devices on your network. That's 6 devices on your network layout plus your other devices, computers, iPhone, iPads, etc. I'm guessing that you might need a router that is cable of providing 25 separate IP addresses or more to allow for future expansion on the network.
So the first question would be......what are the capabilities of the EE router in this regard? Some routers that are provided by Internet Service Providers are set up to only allow 10 devices to connect, for example. Others less, some more.
It really would not make sense to try to set up the EE router to provide all the IP addresses until you know for sure that it is capable of providing at least 20-25 IP addresses, maybe more.
Next item if the EE router is going to provide all the IP addresses..........you have the Time Capsule set up to provide a wireless network.....but.....all of your devices that will have to receive an IP address that is being passed through the Time Capsule to other devices and passed through yet again on some of your other AirPorts to reach a device connected to the network. Makes things a bit more complicated.
If you set up the Time Capsule to pick up only one IP address from the EE router and then provide IP addresses to all the other devices on the network, the Time Capsule can provide up to 200 IP addresses to network devices. You'll never need all that capability, but you'll also never need to wonder whether your not your router in charge will run out of IP addresses.
Another factor to consider is that in an extended network, the other AirPort routers seem to just work better because they are receiving their connection credentials from another Apple router, not a third party router. This makes the set up of other AirPorts easier using AirPort Utility because the set up "wizard" will work better this way.
The downside to setting up the Time Capsule is that you will initially see a message that there is a Double NAT condition on the network, which is really telling you that you have two routers on the network.......the EE router and the Time Capsule router.....both trying to control the network. This is not a problem.....IF.....all of your other network devices connect to the Time Capsule and not to the EE router. Then, you can use the option in AirPort Utility to "ignore" the Double NAT status message and the Time Capsule will display a green light.
So, if the EE router is also providing a wireless network, all your WiFi devices must connect to the Time Capsule. They cannot connect to the EE router in this type of network setup with the Time Capsule in charge of connections to devices on your network.
I think you can easily see why Apple says that you need to set up the Time Capsule to provide IP addresses. I don't always agree with this if the user has a very simple home network. But.......your network is not simple.
If this were my network, I would put the Time Capsule in charge of providing IP addresses. But, this is your network, so it will be your decision on whether you want the EE router "in charge" or the Time Capsule "in charge".
So, we'll need to know how you want to proceed before we start.
In order to check connections as we go.......you really will need a laptop with an Ethernet port to be able to test things......before we connect an AirPort (and set it up again if necessary). The MBP would be great if it has an Ethernet port. If it doesn't.....an adapter to allow this would be well worth the money. It's great item to keep in your desk drawer to test things.
If we cannot test the Ethernet connections of the Ethernet cabling, we'll have to assume that the cable is working correctly. Experience tells me all the time that it is very dangerous to assume anything when you are setting up a network.