Hmm this sounds like a DjSBD sort of question and he's working tonight (its 10pm here in the UK!). Most of the people on here are Mobile D.J's doing your average Wedding, family birthdays etc rather than Mix D.J's or music producers

. There are a few D.J's here who work in clubs or who also scratch etc, so they will probably answer your question better than me

Copyright is a difficult area (and expensive), if you are just messing around mixing in your bedroom for instance, or sharing them with a few friends then I wouldn't worry to much about it yet. However, if you are thinking about selling them on Ebay, or making money from selling a recording of your mixes, then this would be classed as commercial and obviously you would need to be very careful.
For now, if all you want to do is to record your mixes to play them back, or use for demo purposes then I wouldn't worry about the copyright issue just yet.
If you are using turntables to mix / scratch, then you are obviously going to need some sort of set up. Ideally two direct drive turntables (don't use beltdrive for scratching), some form of mixer with a crossfader (ideally user replaceable!) and either an amplifier and speakers or a hi-fi system that has an Auxilary or Mini Disc input so you can connect your mixer to it!. Good quality slipmats and headphones are also required.
That's about as far as my advice goes, hopefully somebody else can advise you on the actual mixing and scratching