I think I see the issue here. Judging by the first screenshot you posted of the DreamPi screen, it seems both the Pi and the Dreamcast have the same IP address. That's no good. lol It will cause a conflict and your Dreamcast isn't going to be able to connect to your network.
To fix this, you need to log into DreamPi with username 'Pi' and password 'raspberry' and then type these two commands, pressing enter after each:
sudo dhclient -r eth0
sudo dhclient eth0
This will force the Pi to get a new IP address from your router.