To answer a couple of your questions:
1) Yes. Idols will wear wigs to hide their new hair color because - new look = next comeback. And they don't want to spoil the look or the fact that a comeback is soon before they announce that it's coming. Both TXT and BTS have had 'wig mishaps' lol
2) The song was a cover of 'Over and Over Again' by Nathan Sykes and it was Taehyun singing. And Nathan Sykes loved it.