"Also, which planes in the Boeing fleet are referred to as "heavies"? Am I right to assume everything above the 737?"
767 and above with the Boeings. It's related to MTOW I recall. A 757 is not "heavy" but "medium" (like a 737).
"Finally, I know that the 737 has a tiller on the captain's side only. What about the 757/767/747/777?"
Company option I think. I know BA 757s and 767s have a tiller on both sides and the handling pilot does the taxy at both ends.
Airbus A320/330 etc have tillers on both sides. Standard Airbus procedure is (quite rightly) for handling pilot on that sector to taxy too. Some airlines who fly Airbus are stuck in Boeing SOPs though and still require the Captain to taxy regardless of who is FP for the sector.