Hmmm.. you have a private IP not public IP on the internet.. I am guessing the building has ethernet distributed for users internet.. but you are already behind a router.
Your current setup will never work.
So I am going to suggest two methods.
1. The TC as you have it now must be setup as router.
And you must move the LAN side to a different IP range.. ie 192.168. You do this under Network Options.
The Network tab should be set to DHCP and NAT.
Your internet tab is probably better set back to DHCP.
This way you will get a double NAT error but that is not relevant.. The ISP is not handing you a public address.. so it won't work like standard setup.
2. The alternative is to separate the Internet from the TC.
Plug the main internet connection directly to the computer. To keep it easy lets say Ethernet 1. Everything on ethernet 1 should be set to auto/dhcp except IPv6 which you set to link local only.
Reset the TC to factory and start over. You must plug it into Ethernet 2.
Leave the wireless on during the setup.. that will help.
You should leave it in router mode (after a reset the TC will be in router mode by default).
You then can manually configure it.
Set the DHCP to 192.168. instead of 10.0 in network options tab.
So the LAN of the TC will take 192.168.1.1 address.
Ignore all the lack of WAN issues.. that won't matter.
Set the computer to static IP 192.168.1.10
Subnet mask 255.255.255.0
Gateway however must match Ethernet 1 and should be
10.10.34.189
For DNS use the same values you have now if they work.. it actually looks wrong to me.
10.18.10.4 is in the wrong address range.
You might find it better to leave it blank and see what happens. Otherwise use the google public.. 8.8.8.8
Anyway I might need a bit more info about the ISP .. and if they have given any info about the setup required.