Setting up TimeCapsule with existing Router(with DHCP disabled)
Disable NAT and DHCP on the router.  Plug the Time Capsule's WAN port into the router's LAN port.  Better yet, see if the router has a "bridge" mode.  The Time Capsule usually would run in 'distribute a range' mode, which IIRC means DHCP and NAT.  'Share a public' would be DHCP only.  And bridge is a simple pass-through mode (much like we've just tried to put your Belkin into).