Yes, it will be compatible with the Kinect, as long as the Kinect was purchased as a stand-alone unit since it comes with the power adapter. The power adapter is needed for the older style 360s, as they do not have the dedicated Kinect port on the back that would typically provide the power.
As for Live Gold, it would be good practice to delete your profile, since somebody else could use it and add to your gamerscore. If you don't care about that and want to leave access to your profile and saved games for whomever you give it to, then just leave it be. You do not have to de-register your account or anything like that, the next person can just create their own Live profile.