Have you tried setting VLC Player's Video/Zoom level to 1:2 Half.
The problem appears to be that your video is larger than your viewing area rather than a problem with the video itself.
When you go to play back your confirmed 1080p60 in the VLC player, do you have a check mark next to Video Tab/ "Always Fit Window"?
I have tried zooming, and i have also tried other media players. I have also uploaded the video to youtube.
The result is still the same, i am missing part of the footage - i find this very confusing.
In your introduction to your post question you specifically mention GoPro 1080p/60fps as your source, and you say that your export from Premiere Elements is a f4v file 1080p. On viewing the screenshots you posted, I assume that the top represents the F4V display in VLC and the bottom one is a view of your source.
That source does not look to me to be 1080p60 (1920 x 1080 @ 60 progressive frames per second). It looks more like 1280 x 960 4:3. Could you confirm on that point? Does GoPro 1920 x 1080 have the same "fisheye" as does its 1280 x 960. Your screenshot seems to be display "fisheye".
The gopro's FOV can be set to narrow, medium and wide. And only when set to medium and wide will you get the fisheye.
I am having my FOV set to narrow, which is why the footage doesn't look "Gopro'ish"