What to do with Time Capsule when not using for Wireless Internet

Question

I previously connected my Time Capsule to my fiber optic router but in my no home this is no longer needed as the cable TV company now provides 160 MPS Internet in a router that contains a wireless router built in. It works fine.  My computer is now connected wirelessly to that router (too far to take advantage of the wired connection).
Therefore I want my Time Capsule to just be connected to the computer for backups.  I gather I have to connect it via Lan and make sure it does not transmit r receuve wirelessly. In fact, other wireless computers in the huse occasionally try to log into the old Time Capsule PPoE settings, so I want to erase them and start fresh.
This seems to be similar to what I want t do, but I want to make sure and ask:
https://discussions.apple.com/message/8256674#8256674
If this is not correct, can someone advise.
By the way, I have a first generation Aiport Express that I hope to use with my new cable router (modem?).  I assume it will work as it did with the Time Capsule. Purpose is for Airplay with an audio system.
Thanks!

Answer 1

Thank you for your replies. I really appreciate it.
To answer your questions in order, but first I should mention that due to work reasons, I will be away from my computer until Friday night Japan time, or Friday AM US time.
Anyway, probably better I go slowly,as I tend to get overwhelmed by technical things.
1) You said:  "The TC does NOT require an Internet connection for network connectivity. You DO have to connect the TC by Ethernet to the Fiber Optic router in order for it to gain network connectivity AND to allow network clients to access it."
-->Ok, this is in reference to my Time Capsule and the previous network n myold apartment.  But your next question makes me wonder:
2) "Is this Motorola device the one that replaced the Fiber Optic router that you mentioned in your original post? What is its model number? Do you have the TC connected directly to it by Ethernet?"
Ok, this is what I have:
Wall--> Cable modem with wireless (Motorola SURFboard SBG6580J Wireless Cable Modem Gateway.)
Other Room  IMAC connected wirelessly to Cable Modem (no Ethenet connection to it since there is no jack and too far to run a cable (permanently--I can do for setup if needed)---> connected to TC and Time Capsule is backing up ok. Time Capsule is also not connected to the Modem
3) "Since you will only want to use the Time Capsule for Time Machine backups, I understood that you wanted to disable its wireless network. If this is still the case, the Express would need to be configured to join the wireless network provided by the Motorola wireless router."
Yes, This is what I am wondering. Do I have to connect it by LAN to the Motorola?  A few times, when it was just plugged into the Wall, my Mac airport utility found but them could not connect to my Express. maybe because I did not configure it to work with the Motorola as you say.
4)
"By default the Express will broadcast its own wireless network with a network name of something like: Apple Network NNNNNN. You would need to connect to this network with your computer prior to attempting to access it with the AirPort Utility. Another option would be to temporarily connect it, by Ethernet, to the Motorola router during its setup phase. Once completed you can move it to the desired location."
Ok, this confirms what I guessed.  Connecting the computer to the Expressby Lan?  Is that what you mean?
I do understand the second option---connecting the Express to the Motorola. 
If that is right, I can try that.  Do I need to do anything other than connect it?
I will be able to try this Friday or Saturday I hope.
Once again. thanks for all your help. Really appreciate your taking the time to explain it.
paul