To unlock World 11, beat Worlds 1 through 10. Then, go to Hollow Diamond, turn 180 degrees and go slowly up the ramp. Land on the narrow platform below, then turn around again and hit the red gate. Go to Converse, then move towards the right edge. Wait for the red platform to pass its vertical angle. Gain some speed and dash at the red gate and World 11 will become unlocked.
On the "Start Game" screen where AiAi is dancing, there are flowers under him. Tap them, and they will dance. If you tap birds, they will fly to the top of the screen. Tap a balloon two or three times to make it pop.