First of all, to turn off "Dial-Up Pop-Ups", go to Internet Explorer, then Tools--> Internet Options--> Connections, and click on "Never dial a connection" ratio button.
Second, You should understand that you have different naming conventions andmechanism in "Microsoft Networking" (used for file sharing and neighboors browsing), and legacy TCP/IP applications (like ping, ftp, telnet, etc.).
1. "Microsoft Networking" uses NetBIOS, names can contain space and can be long. Windows uses broadcasts to find NetBIOS names in the network, so, it will work without any configuration required.
2. TCP/IP uses more conservative names. You can't use certain symbols in names ("Space" is one of them). IP uses 2 mechanisms to resolve names - local "hosts" file or DNS. And when you try to use name in "ping" command, system checks local "hosts" file to get IP address of workstaion you are trying to reach, and since it's not there, it try to dial to ISP to get address from them.
The solutionis simple. First of all I advice to configure your workstations with static IP addresses, since you have only 2 PCs. Second, you should edit c:\windows\hosts file to configure name-to-address resolution. You may just create file and copy it to all PCs.
If you will have problem performing these steps, don't hesitate to send me e-mail.
Good luck,