As of 15th June, 2015 yes you can! Microsoft announced at the e3 conference that xb360 games WILL be playable. (this list is small but is expected to grow once all game makers sign an agreement form). Also you wont have to pay to play games you already own on disc or digital.
By Rajiv B. on June 17, 2015
Kinect is for Motion Gaming, using Gestures to control the Xbox, and most importantly voice commands. Sadly no, a different external microphone will not work for voice commands. Head over to feedback.xbox.com and suggest it, maybe they will add it.
By EdO on June 10, 2014
Most games do not require kinect. Some games have certain kinect-enabled features and others will rely on kinect extensively. It will be marked on the game packaging if kinect is required.
By M. Paulsen on June 12, 2014