Just wondering what kind of band width is needed for mining.
Not much - I was on a 12Mbp/s download speed sat connection with a 10GB/month cap and never came close to the cap from mining - I think I MIGHT have been going through as much as 1 Gig/month but that was with up to 14 ASIC devices AND 7 or 8 Ethereum-mining computers running at the same time.
I also mined for a while on a 3G Cell connection (was LUCKY when I saw better than 400kb/s DOWNLOAD speeds on that thing) - no issues, and lower reject rate as the lag was a good bit lower most of the time.
I suspect your "lag" issue is something else. I used to see my reject rate go WAY up when I was downloading a large game patch, for example (World of Tanks was particularly bad about that).