I store mine in these acrylic desk tidies, if you buy the sets these are what they come in. I myself just buy the pens seperately so bought these for them to be stored in. I got mine from All That Scraps in Canada and Cult Pens here in the UK which is where I buy the majority of my Copic pens too. I try and store them in their colour families so it easier for me to find when colouring an image.
In terms of colouring I use a thick cardstock - mostly the Softies range from Craftwork Cards. As this is so thick it will dry my pens out quicker than using thinner cardstock but persoanlly I like this cardstock and it works well for me. I would love to know what card you like working with using Copics? I always stamp with Memento Tuxedo Black as this has never run for me. I used to use Airondack Pitch Black but I did find that it did run on certain cardstocks. I definately wouldn't recomment Stazon though for Copics, that defiantely does run.