Yes, they can definately sting when still dead, the nematocysts (the bit that stings you) works in a mechanical and chemical way, so touch fires them, it is not done under the control of the jellyfish. also if you have been stung do not rub it with your hand as there will also be nematocysts left on your skin which will fire both into your hand and more into your sting.
It is also not true that peeing on it will always help. for one thing it will not help the pain at all. Vinegar is good for box jelly stings as it hardens the nematocysts left on your wound and lidocaine antiseptic spray is good for stings from things such as man o war and sea nettles. treatment for jelly stings is very much species specific.