Welcome to Amiga.org! :pint:
You've already gotten a nice collection there! Pity about the DOA 1084 though :-(
As far as getting your A1200 on the net, it's really not fast enough. You can definately get it on a network (you'll need a TCP/IP stack, try Genesis or Miami), but without some more upgrading, it's really not fast enough for graphic web browsing. Text based should be fine, but for graphic browsing at an acceptable rate, you're really looking at at least a 68040 accelerator card. Realistically, you'd want a 68060 and an RTG graphics card (for which you'd have to tower it.) Basically, it's more than fast enough for networking, but for net browsing, the AGA chipset isn't fast enough. You can always give it a go, but I wouldn't attempt it without at least an 030 accelerator. If you're looking for good accelerators, check out a Blizzard 1230 or a Blizzard 1260.
Hope that helps. :-)