Congrats on setting up your new system!
Networking - how can I get it on a network? Back when I used AmigaOS I didn't have internet. Do I have to get a network card or can I use some sort of serial cable and bridge it with my desktop?
Serial links can be done, but they're hard to configure and sloooooow. Best grab one of
these instead.
DVD Drive - I have a SCSI DVD-rom. Does OS 3.1 have CD support or not?
Kinda sorta just barely. It exists, but it's really lousy. It'll do if you need to sneakernet some files over from a Unix box, but it's best to keep them archived and then extract on the Amiga to preserve the correct protection bits (and filenames - OS3.1's CDFileSystem makes a mess of them). AmiCDFS, AsimCDFS, CacheCDFS, and BabelCDFS are your other options. The fact that you're using SCSI makes things easier - no need to mess around with ATAPI drivers.