If two computers connect to the same domain name, do they share an internet connection and network?

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Depends on your network setup. The domain name has very little to do with this; just because myself and my neighbors are on Comcast.net as a domain, we don't share the same internet connection. What determines if you're on the same network has more to do with the Subnet Mask and IP address range configuration of your DHCP server.

For example, if one computer was given the IP address 192.168.1.100 and another computer was given the IP address 192.168.1.101, and both were on Subnet Mask 255.255.255.0 they would be able to easily talk to each other on the network without any routers. Both computers would also share the same internet connection.

On the other hand, if one computer had the IP address of 10.0.0.5 with a subnet mask of 255.255.255.254, and another computer had the IP address of 10.0.1.6 with a subnet mask of 255.255.255.0, you would need a router to allow those two computers to talk. Both computers would still share the same connection to the internet, however.

So the short answer is determine the Subnet Masks and IP address range of the two computers, and that will let you know if they are ready to talk to each other or not. If so, configure file sharing, printer sharing, and stuff like that, and have fun.