In levels with Tails, you press the Y button to call him to you, similar to how the elemental shields were shown with Y in that one screenshot. It’s likely all those little powerup icons we saw are things you can use with Y.
You can call Espio in a fashion similar to Tails. He has a grappling hook.
There’s also the ability to call Vector. He throws you to locations.
A team-up mission with Amy is included; she gives you the hammer jump.
There’s a team-up mission with Cream in a level with no rings. The only way to get rings is to call on Cream to throw you some.
There’s a treasure hunt for buried metals. You call Knuckles, who digs into the the ground for you.
Rouge team-up: heavily armored robots block you–call on Rouge to take them out.
Blaze team-up: the level has walls of flame which Blaze can extinguish.
Charmy team-up: …he can make a vortex at your feet. What?
The Tails race is with Tails in a plane.
In the race with Knuckles, he will burrow underground to chea–er, use his unique abilities to take shortcuts.
Rouge tries to stop you with bombs in her race.
There’s an Espio fight in which you have to shine a spotlight to find him while he’s invisible.
“Collect 10 Chao and reach the goal before Cream!” (…explains itself.)
There are time trials that have powerups that give you more time.
The doppleganger race is not against the other era version; it’s essentially a ghost race, complete with translucency.
In the challenges with the giant enemies you just knock them over instead of killing them, but then they can be used as platforms.
The icon with the motobug and the ring is for Ring Time, which changes all enemies into rings (presumably for just a few seconds.)
There are apparently challenge missions with multiple animal capsules, where Sonic has to release all of them.
There’s a mission where Sonic has to carry the sign post to the end and knock it around, presumably like the end of Sonic 3 Act 1s where you could juggle it. The player has to knock it to the end of the level without it falling into pits.
There are single-ring challenge missions.
Jump panels are in at least one challenge act, presumably as in Sonic Adventure.
There’s a time trial in which your boost meter is 3 times the size of normal but it doesn’t replenish. That explains the “300% Boost” seen earlier.