Yes, because "playing tennis" uses the gerund form. So you treat the noun-clause as a thing, not an action. "What kind of things are you into?" "I am into ______"

Yep. Perfectly correct.

"I am into play-ing tennis" should be right... However, saying " I am into playing tennis" is more of a 'slangy' version of saying " I like to play tennis a lot" or something similar..... so, it might not be the best choice if you're writing an essay... =D I might be wrong though..... by the way, Roger Federer!

Yes, it is correct. Either "I am into playing tennis" or "I am into tennis"... both forms correct.