I'll answer this in stages:

1. Yes, you can change your videos to mode 777 if you would like. That will not affect the script.

2. If you are wanting to edit a video file and cannot overwrite the video via FTP, then changing the file to 777 so that you can do so is the wrong approach. What this means is that your web server is writing all of your files as a user other than the one you are logging into your FTP with. I would (from a security stand point) look into changing the user that Apache is using to write your files as versus changing the mode to 777.