Translation:Would you like to play basketball now?
打 literally means "hit". You can see this in phrases such as 打字 = "to type" (literally "to hit characters" as in pressing keys on a keyboard) 打电话 = "to call on the phone" (literally "to hit the phone" as in pressing the numbers)
踢 literally means "kick".
So you 打 sports like basketball and pingpong, but you 踢 sports like soccer.
99% of the time you can use 'want' or 'would like' for 想. But every now and then the answers select one or the other and insists that is the correct meaning. Drives you nuts, and totally unpredictable.
Of course 想 is a more polite and less wilful way of saying 要, but as I say, they use either character to mean 'want' when it suits them.
My Answer to this question: "Do you want to play soccer now?"
No it has to be "Would you like to play basketball now?"
Irritating as usual.
No that's all of a sudden wrong.
This signifies giving the listener an option to do something. Literally it means "do you want to play basketball now or not" but doesn't have the aggressive / demanding tone that the English sentence might have. This is probably the reason why it's not translated that way as it can be misconstrued.