Full Screen mode is too wide
 

Whenever I go to full screen mode on a video whose relative width is greater than the screen, the video is enlarged until the height matches, meaning I lose the edges of the video. This doesn't happen when the video is too tall, only too wide.

are you sure that you don't have pan&scan enabled ?
(right-click on the video -> video -> pan-scan

[well pan-scan is not enabled by default (I mean whenever I close/reopen bsplayer, pan-scan is disabled) so this sounds strange...]

Check if you have enabled 'Cut top and bottom of video in full screen mode' option. Under Preferences->Video->General