Yes..
Last question first.
How do I check whether " the building network is based on 10.x.x.x"?
The airport utility will show you what IP you get on the Time Capsule.
This will be easier (IMHO) to do on your laptop version airport utility.
Click the icon of the TC. I am running both a gen4 and gen5 but I can easily mess up the Gen4.
It is currently in bridge mode so it shows the same IP on both IP address and LAN IP address. If that happens to be 10.x.x.x where x is any number between 0 and 255 you will need to move the IP of the TC. If it is anything else you don't need to worry.
To change to DHCP and NAT.
Go to the network tab.. select choice of router mode..
Click on update.
You should then get an error.
This is double NAT.
And the fix is to ignore it.
You suggest I should change to DHCP and NAT - but then I'll get another error, that I can't do anything about?
Well it is not really a fix.. but that is required because your building is not passing you a proper public IP. This is fairly standard in hotels and apartment buildings which supply internet.
It will make very little difference to your network usage.. and will block other people in the building accessing your clients. Otherwise they can easily access whatever you connect.
Now is it all happy again.. but it is using double NAT.. which is Apple's solution as well btw.