First of all, don't edit with H264 it's a delivery codec and FCP will give youproblems. Convert your iphone footage to pro res and edit with that, then export as Quicktime Movie, not Quicktime conversion, and then use Compressor to export to your delivery format whatever that is.
Regards your lines in the video, it looks like interlacing to me.
More than likely your sequence settings are interlaced because your H264 is interlaced.
You could try duplicating your sequence and changing the sequence interlace setting to none(click seq/command+zero) and then render. But H264 files in FC7 do not place nice.
Can you do it anyway? yes
Are there numerous posts similar to David's suggesting you don't? yes
Are you almost guaranteed to have problems? yes
Panning can also cause interlacing issues as the camera can have a hard time keeping up with the motion depending on the frame rate of the movies you are shooting.