Should I clean all of my cooking utensils after being exposed to mothballs (naphthalene) for over a week?

I doubt that there's enough naphthalene deposited on the cutlery to give you severe poisoning, although it may taste bad. Based on the rat/mouse lethal doses, you would need to ingest ~50 gr of the stuff to die from it (of course, you would experience adverse effects at much smaller amounts). It seems like chronic exposure to naphthalene is where you really start getting problems - so definitely don't mothball cutlery as a regular thing in the future! Nevertheless, it is a harmful chemical so I would wash them just to be sure.

Naphthalene is an organic compound and has very poor solubility in water (32 mg/l at room temperature) so regular washing might not remove it effectively. Meanwhile, alcohol can dissolve 11.3 g/l (1000 times more!) and acetic acid (vinegar) can do 13.1 g/l. Granted, soap is supposed to solubilize things, and since you would wash with soap and not just water, solubility shouldn't be as big a problem in reality, but it's worth taking care to ensure it actually gets removed. So my needlessly complicated procedure that I would suggest to you is:

• Thoroughly wash with vinegar, as warm as possible (use rubber gloves)
• Thoroughly wash with soapy water, as warm as possible. Remember to also wash the gloves and your sink to remove residue on them.
• Thoroughly wipe with alcohol. I think ideally you want close to 100% (if you have Everclear that would be about perfect), but in a pinch rubbing alcohol (typically 60-70%) would do or even a strong spirit like vodka. After wiping the alcohol off, leave to dry as alcohol will quickly evaporate.

At the end, ventilate the room, wait an hour or so, and see if you smell any of it from sink/cutlery. Hopefully the answer is no and you'll be good to go. This method is probably overkill and will REALLY get the naphthalene out, but it would only take a few minutes (excluding the drying and checking for smell) and you don't have to worry afterwards.

For cutlery, I don't think it's worth throwing them out for something like this. Naphthalene is pretty volatile, so even leaving them in a warm, well-ventilated place for a few days without the above washing would probably get rid of most of it as the napthalene evaporates (technically, sublimates) and dissipates in the air. With pans, I'm not so sure. My guess that there's no difference and the pans can be used after cleaning just like cutlery, but then since you heat those, small amounts of it conceivably might react with the teflon or itself and make something nasty. I would throw away if it's a very cheap pan, if not just wash with vinegar, soap and alcohol as I say above and you should be fine.