1. Agreeing with a two year old is better than asking questions. If you ask questions, they answer and the game stops. Example. Do you want Elmo to go to the park?
Child will say yes or no and then what.
If however you say, "I'm taking Elmo to the park"
The kid says,"Let's take him on the swings."
Now, you've got a narrative.
2. The "game" in improv world is what happens that is unique, strange, or remarkable about the interaction between the people on stage. Example...a senior partner at a law firm is so arrogant that he can't even compliment a junior partner without it being a put down. He just gets worse and the junior partner quits the firm.
With a child, the game is less complex. Examples....giving Elmo and non-branded dolls a bath. The game....shampoo in their eyes...they either get shampoo in their eyes and they cry or you are careful to keep shampoo out of their eyes. That is the centerpiece of each bath scene.
3. Props can work. My granddaughter (2 1/2) picked up a ukulele. I suggested that she play "Twinkle, Twinkle little star." She played that. Then she became a song leader and made up words to the tune of "Twinkle, twinkle" and had the doll students standing up, sitting down, dancing and singing.
FYI...there 57 million google results for Elmo and and 6.5 million image results.
Here is Joe Pesci, playing with Elmo,