How to fix FLV files: Remux FLV (h.264) file into MP4 file
You can remux an FLV file into an MP4 file quickly without any transcoding.

Get freeware FLV Extract and freeware mp4box.

Using FLV Extract, drag the recorded/downloaded .FLV file to the FLV Extract window. It will demux into video and audio files, ending in .264 and .aac

Then, use mp4box to remux those files into an MP4:
Where "filename" is the name of the .264 and .aac files. The output .mp4 will play properly and have the correct time length.

Instead of mp4box you can use freeware GUI utility YAMB.

Please note that mp4box and YAMB may give different results