Most people use either IIS6 or Apache/Tomcat web servers. For sake of configuration, probably should get Windows 2003 web edition (don’t forget to install IIS lockdown unless you use application filtering on your firewall).
As for DNS, just register your Domain Name with someone like Network Solutions, they will host your domain DNS servers so you can easily add, remove, or change DNS entries. After you register your Domain Name you can use their DNS tools to create NS records to point the domain name to your public IP address, then of course setup your firewall to NAT to your webserver.
The network connection that you need will depend on the amount of traffic that your web site gets. If you will just have a few users at a time then a T1 will probably be fine. If you need to handle hundres of users at once then you may need a larger connection. Also the size of the web pages which you’ll be serving will adjust this. If you have a lot of flash, or streaming videos then you’ll need a much faster internet connection.