Question: I've heard and read about how recording/mixing engineers in the music business analyze each others mixes to learn the techniques used to make a certain track sound so good or what the latest trend/style is being used for them to incorporate in their work. They listen to how the track is balanced, panned and where each instrument fits in the frequency spectrum and so on, I'm guessing the same applies to a film or video game?, what else is there to look for?

Answer: 
*light spoilers ahead for We Own the Night, Inglorious Basterds and Michael Clayton ahead*

Be careful not to get too lost in the technical details. Be mindful of the overall purpose of the sound design. I brought this up in another question a couple years ago... the movie, We Own the Night, starring Joaquin Phoenix and Mark Wahlberg has a very compelling car chase sequence in the rain where the viewer gets the sense of claustrophobia as the unwilling passenger in the car that's being chased thanks to great editing and sound design.

In Inglorious Basterds, the scene where the Nazi officer gets beaten to death with a bat is pretty intense. You get this great buildup as Sgt. Donny Donowitz beats his bat against the wall of this long tunnel for an eternity... before he even appears on screen. Accompanying his entrance is this epic music that heralds the Bear Jew and just when the tension and music builds up to its highest point, the music is pulled out and all we hear are the cheers of the American Soldiers and the brutal pummeling of the twitching Nazi officer. It kind of makes for a rather dystopic experience if your sensitive to such brutality. At the same time, the movie does not really tell you how to feel about the scene. It's pretty much determined by how you feel about the situation as it unfolded.

Lastly, there's a scene in Michael Clayton near the end where Michael's Car blows up in the background. You see his car in the distance and out of focus blow up and almost a second later you hear the explosion. Perhaps someone accidentally slipped the explosion SFX behind a second or so by accident. Or, was it done on purpose to create a surreal effect. As if to ask yourself, "Did my car just blow up?" only to get the affirmative answer a half second later with the explosion. Or perhaps that happened as recorded because light travels faster than sound and to get that delay, Michael was far enough away from the effect to watch that delay occur (*shrug*) 

Anyway, my point is to also consider the context or the role sound design plays in supporting or telling the story.
I found it effective to use Adobe Audition and XMind (mind mapping tool). 

In Audition I am able to see both waveform and spectrum at the same time. I set markers at the in and out points of certain audio sections that I'm interested in and then use Marque Tool to listen to exact frequency range of the sound. I can reverse, de-noise, make it louder. 

In XMind I have a template mindmap for this kind of task with several sections that I need to fill in, like Melody, Harmony, Rhythm, Instruments, FX, Structure, Mix, Automation, Stats etc. I tried to use simple lists and spreadsheets for this but I'm a visual person and mindmaps just work better for me.