An A2000 certainly is internet capable. I have an A2000 with a 2060 Blizzard, PicassoIV, X-Surf and 100 MB Ram. It works quite well speedwise although it is sluggish compared to my microA1. The only limitation is Amiga browsers in general - and this applies to all Amigas.
I have also used '040 and '030 cards in my A2000. The '040 would be usable but I think the '030 would be too slow in my opinion.
