Time Capsule network, Windows machine on different subnet?
If your Dell running Windows 7 is connected via ethernet or WiFi directly to the Time Capsule, then it sure should be able to ping. Perhaps a firewall is blocking those packets? Also - you might want to manually assign a 172.16.x.x address to be sure it's not a routing issue.

Airport Utility uses IPv6 when it can as well as Bonjour (mdns) so it will find attached units more aggressively than ping. Also, I've connected to Time Capsule from PC across the internet, so it's not something that has to be on the same subnet - just that the networks have to be working correctly.

In your case, 255.255.252.0 isn't a wide enough netmask to unify 172.16.x.x with 192.168.x.x. However, shares work over the internet, so it's really a routing issue and not something necessarily wrong with a netmask. Can you at least get all the machines on the same subnet?

Mobile Me also solves this problem for mac users but I doubt that's something you can sign up for during the transition to iCloud (and it would merely work around your network issues, not solve them).