Definitely do Elephant Mountain instead of going up Taipei 101. The view is so much more beautiful WITH the 101.
Shilin vs. Yongkang: I'd say go with Shilin (but not if you have your luggage!). There are snack vendors and clothing stores and stalls and everything in between. Yongkang Street is just a neighborhood with mostly sit-down restaurants. It's a completely different feel.
Din Tai Feng - I'd say no. The food is nice to marvel at and to eat, but for the price and the experience, it's not really worth it. Try Shin Yeh or Yinye instead (: