Do you need to do the online challenges to complete the game at 101%? Or do I just need to complete everything eles?

Yup, online challenges only.

Completing all of single player eanrs you 101%, and doing all of the challenges as well will give you 102%.

I thought the 102% was for beating a set of the new timed challenges on top of the ones that came with the game.

100% - Single player completion (elite license, road rules, and all discoverables)
+1% - Complete two full blocks of Freeburn challenges (example - all 50 3 player challenges and all 50 6 player challenges)
+1% - Complete all 70 Timed challenges from the Cagney update