Convert 360 to flat video
Yes, I believe the latest versions of Premiere CC can do this natively.  It's a little bit tricky since the video needs to be unwrapped, but it is a fixed distortion to be applied depending on the size of the sphere map.  360 videos are actually rectangular videos that are texture mapped on to a sphere.

Software that wants to convert can either look at the size of the sphere and use a fixed distortion to get the correct output or actually execute the spherical mapping and then shoot a flat image of the mapping in a 3d translation.  The later approach is more complicated by a bit, but allows for easy manipulation of the camera within the 360 sphere.

I'm not particularly aware of any other applications that directly support working with 360 video yet.  I know Fusion has VR capabilities now, but have not had a chance to work with the new features yet.