Howdy Mr King! As has been said, really you should be looking at going to a specialist running shop (not somewhere like JJB etc...), whilst you obviously have an idea of what you want from your trainer, it unfortunately doesn't mean that any of the trainers you have looked at will suit your running style.
Why does it matter? Not all that many runners have a perfect stride, a specialist running shop will be able to see how you run and what level of support, cushioning and/or motion control you require - ultimately helping to stop you having to posting on here saying how you've developed pain in your shins/knees/calves
As for weightlifting, I'd just use some normal sports shoes rather than running shoes in the gym as the running shoes tend to have a large chunk of cushioning on the bottom. If budget doesn't allow that then you should be okay with most trainers, just avoid ones with things that would have the ability to break (such as air pockets, 'shocks' etc).
Hope that helps a little
