How many airport routers can be on a network?

The first TC is a the main router for wifi and wired network and Time Machine back up.

What exact model number is your TC? Actually, what are the exact models of each of your AirPort base stations?

The second capsule doesn't do any routing not expanding the wifi network. Just being used as a second time machine back up drive.

How is this second TC connected to the network ... by wire or by wireless? Is it configured as a bridge?

The express for airplay, no routing.

Is the Express then configured to join the wireless network provided by the first TC? ... or is it connected by wire to either of the other TCs?

The extreme is to expand the wifi network but, is linked using ethernet.

Since it is "linked using ethernet" is it configured as a roaming network with the first TC? How did you try to use the "extend a wireless network" option when you set it up?

My thought is that I have too many airports on the network or one of them in conflicting with the main router.

Possibly, but most likely, you may not have them all configured properly to meet your overall networking goals ... which would help if you can tell us what those are for these devices.

How many airport routers can be on a network?

It all depends.....on whether the AirPorts are connecting to the network using a wired Ethernet cable connection, or whether the AirPorts are connecting to the network using a wireless connection.

If the AirPorts all connect using a wired Ethernet cable connection, then you can have as many AirPorts as you want..... within reason on a home network. That being said, I've seen networks with 20 AirPorts and things work fine.

If the AirPorts connect using wireless, you already know that each AirPort must be located where it can receive a strong wireless connection. And....if the AirPort is set up to "extend a wireless network", it must be located where it can connect directly to the "main" AirPort for the network.....since Apple will only allow you to extend the network one time.

You can have more than one AirPort extending the network, but they must all be located about the same distance from the "main" AirPort.....and since they must all connect directly to the main AirPort, all of the extending AirPorts need to be closer to the main AirPort than they are to each other.

Extending a network using a wireless connection will always result in a loss of about half of the potential speed capability on the network, so it would be wise to limit the number of AirPorts that connect and extend the network using wireless to as few as possible.

If the AirPort merely connects or joins the wireless, and does not extend it, then you can have as many as you want......within reason on a home network......keeping in mind that each wireless client will slow down the network a bit when it joins the network.

I added a airport extreme to the network and now the wifi fails and the time capsules and the extreme won't show up in the airport app

This can happen if the AirPort Extreme was originally set up to "extend a wireless network" and an Ethernet connection was applied to the AirPort. A huge feedback loop is created on the network when this occurs, effectively crashing the entire network.

The solution is to start over and reset the AirPort back to factory default settings....then make the Ethernet connection first..... before.....you set up the AirPort. That way, the setup "wizard" will recognize the wired connection and set up the AirPort correctly to work on a wired connection.....not wireless.

Your plan to troubleshoot the network by starting with just the Time Capsule is a good one. Make sure everything is working with the "main" Time Capsule, then add the next AirPort device and check again. Then, add the next, etc.

If you still need more help, we'll need specifics about your network.....what connects to what.....how it connects....etc, basically the whole layout of the network in detail.

I think what happened was that I didn't reset the extreme back to factory settings

When you said that you......added a airport extreme to the network and now the wifi fails......we may have made the dumb mistake of assuming that this was a new AirPort Extreme.....with factory default settings.

If it was not, then you would always want to reset the AirPort back to default settings first.....before....you add it to a network.

If you are saying that you reset the AirPort back to factory default settings and it is still crashing the network.....then you likely have a defective AirPort.

But, to make sure, we need for you to tell us exactly what steps you are using to set up the AirPort Extreme to the network.