I wear corss trainers ONLY for weights. You should not wear running shoes for lifting, especially legs. Running shoes have a "lift" in the back, the padding, and can throw off your stance - like on squats. It throws your body toward the toes. Also, there is NO lateral support, so when doing any movements, you are at risk for twisting an ankle or at least, feeling unstable.
Nike Shox are normally running shoes. Cross trainers are typically flatter and better suited for weights.
I wear running shoes for most cardio - but not all. On the elliptical running shoes put too much pressure on the balls of my feet, so I wear cross trainers there too. ALso - for classes, such as kickboxing. Again, you do not want to wear running shoes for kickboxing (or most other cardio classes) - the lack of lateral support can kill an ankle.