No, you don't HAVE to have a laptop when DJing but, given that most DJs tend to now download their audio files, it does make life a little easier to work from the laptop. Think of it as an electronic way of flipping through your CD wallet.
Depending on the DVS software you use, you will see that pressing play on the CDJ350 will engage play on the software - the CDJ is acting as a 'controller'.
The benefits of using a DVS with the CDJs are that you have multiple formats to play music from - a laptop / USB stick or CD. You can access your entire library from the software and then the software plays that music.
I do use a mixture of setups depending on the gig I am doing. If its a mobile gig I tend to go for the laptop approach as I have faster access to a wider variety of music than I might have on some USB sticks (for example, I was playing a Rihanna song last night when I got a request for Garth Brooks - there is no way I would have had those tracks on the same CD or USB stick).
Its not quite plug and play but its getting there. You have to do a certain amount of 'mapping', that is, to assign specific buttons on your CDJ to control specific functions within the software. You can download mapped templates. Do a search on these forums for some .tsi files.