Hmm, I am trying to work this out. Please tell me if I am correct -
Your fish died so you emptied everything out of the tank - did you get the water tested before you emptied the tank? If so what were the readings.
You then cleaned out the tank, and put in new filter sponges & carbon, filled it with tap water, waited a week (and in this time added nothing to the tank, absolutely nothing - no fish, fish food, additives) and then took a water sample in to the shop and it came back with .5 for ammonia, .5 for nitrite and 10 for nitrate?
You have no plants, driftwood or anything else in the tank? If so were they in the tank at the time of your fish deaths?
If the above is correct what are your readings straight from the tap - are they the same as the readings you got when you took the water from the tank in to the shop?
Just an aside - carbon is chemical filtration - best used when wanting to 'polish' your water or remove medications ie the blue from meth blue. Carbon needs to be replaced regularly as when it is 'full' it will release the waste and toxins back into your tank.