Having a music background and DJing
In response to the question: Does it make a difference if someone has a musical background if they want to become a DJ?
Personally, I don't know if having many years of music background is making a difference when it comes to mixing. I know some DJ's who sound fine and don't know a thing about traditional music theory. I also know that mixing is something most acoustic musical performers are not good at.
Now, I don't think I can tweak (at least well) things like mixer levels, frequency balance, and FX parameters. Understanding the fine points of a low-pass EQ filter (resonance/gain for example) or using saturators is not something I was taught when learning piano... let alone song transitioning. I think it's a huge shame when DJ's mess up beat-matching at shows. The vibe is just shot for a little while - if I have options, I might just leave the room. Maybe it's because I'm approaching this from my dancing perspective... I also have been spoiled by software algorithms that do lots of heavy lifting.
A little bit about my perspective on this... when I used to "DJ" for house parties (top 40/mashup), I used Live 6/7 and spent 20% of the time actually thinking about what songs to play and 80% learning how to mix random songs together... the reason being that pre-conceived song order/mixing in a live gig only goes so far when your audience can come right up to your setup... I liked taking requests on the fly or reading the crowd, pulling in whatever came to mind, and trying to mix new tunes into a set. Sure, I would MAYBE scribble down the key, but I was more interested in putting in markers (especially for acapella tracks, which can be a nightmare to beat-match quickly) on key transition points in the song. Mixing knowledge/skills were so much more valuable to me at that point than knowing song 1 was a half-step up compared to song 2 or song 1's C7 chord might clash with song 2's F#m... not to say that wasn't important, just not a priority. Maybe it's a bit more relevant in trance music - not a huge trance person...
I always told other DJ's to practice beat-matching/mixing skills first, and if they were interested in acoustic music to learn how to play percussion/drums. Nowadays, I'd probably tell someone to listen to their favorite DJ's mixing and try to replicate those mixes and just ignore the whole drumming thing. But I'm not a regularly gigging laptop mixer anymore, so what do I really know about this...