Something simple as Audacity allows you to cut up and edit tracks. Obviously any serious DAW will let you do the same, but at a way higher price tag.
You can make mashup with anything that allows you to mix parts of tracks, so yes, you could make mashups with your DJ software. Again this can be done in a DAW, stuff like Ableton, or specific software like Mashup by Mixed in Key (even an iPad version available, I think, so you can play around while on the road).
Sure some of the other readers have useful comments to add to mine.
Hope this helps some.
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