The games themselves account for Emerald progress.I feel like I need to stress it again because NO English media mentions it, but there is a Story Mode that goes through Sonic 1 > CD > 2 > 3&K (in that order) together as a single campaign, and the new animated cutscenes are for THAT mode. It's the first option on this Spinball inspired menu below:
No indication of how character choice or Emerald progress carries between games from the description.
Site also mentions that some of the purchasable items for the museum are "never-before seen" materials from the original game development.
At the end of Sonic 1 the 6 emeralds Sonic collected make some flowers bloom and then fly off leaving Sonic confused
At the beginning of Sonic 3 Sonic still has the emeralds from Sonic 2 and Knuckles punches them out of him and steals them.
At the start of Mushroom Hill Sonic warps to Hidden Palace Zone and the emeralds leave his possession and turn into the Super Emeralds.
The tricky part will be what happens if you don't have all the emeralds at the end of Sonic 1 and 2? Will Sonic start Sonic 3 on a surfboard if you don't have all the emeralds from Sonic 2?