are you sure ? Samsung devices require that you select the mode before connecting the cable, after which you need to confirm USB storage mode (as others have mentioned).
As far as the wifi probs are concerned, are you using a Dlink router ? Have you tried a different wifi network and get the same wifi probs? If yes then you might have a faulty device ? I've not checked to see if this is a known fault with the SGS3. My Nook Color has the exact same wifi probs with my Dlink router but works fine on every other network. I added a Belkin wireless gateway to my network and the problem disappeared.