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.