To unlock Future World II, you have to collect a minimum of 405 Sparks. If you collect all Sparks from all Challenges and Create Chains as you go, you can unlock Future World II as early as the Darkworld (World 10).
I have recorded my own solutions for all Challenges from all 14 Worlds in the game. Following is a series of videos of all challenges from Worlds 1 to 13.
These were recorded after I completed the game. They do not include the Sparks earned from Create Chains.
Prior to the servers closing, you could use an in-game gallery to view other people's solutions. However, since these servers are now closed the only way to view other people's solutions is via YouTube videos. If my video solutions do not work for you, then I'm sure other people's will.
Please either comment below or send me a Personal Message if you have any comments or improvements.