It is now possible to move items behind your pet. Just tick the "rearrange mode" tickbox at the bottom right of the screen, drop an item onto the pet, click the item, then click the little down arrow which appears to move it behind your pet.
"Rearrange mode" is now also available on Internet Explorer (previously you could only use it on other browsers, like Firefox or Chrome).
We're going to be converting some of our older "background" items to fill in the dog-outline shaped holes, so that you can be more flexible with the positioning of those
Dressing up your pet should now work on the mobile edition, but I've only tried it out on Android so far. It's a little tricky to drag items which I hope I can improve a bit.