This is a funny thread... I belong to many automotive detailing sites (I manufacture high end detailing products, www.dftowel.com) and whenever the subject of cleaning detailing towels comes up the thread explodes. Look at all the responses here! There seems to be something about iPhones, Hot Rods, and laundry!
Anyway, the cloth that comes with the phone is far from magical, it's a simple polyester cloth. The one thing you DON'T want to do is clean the phone dry. All you will succeed in eventually doing is catching a piece of grit and grinding into the surface causing scratches. Over time you will see a lot of swirl marks. What you want to do is just spray the cloth, not the phone, with a VERY LIGHT spray of glass cleaner then clean the phone.
As far as cleaning the cloth is concerned, just throw it in the wash with your other darks and let it go through the cycle. Machine drying won't harm it at all. You want to machine wash the cloth because hand washing will not always get out all the dirt and oils and will eventually contribute to scratching the phone.
You can eventually replace the cloth with eyeglass ones from an optician or use a small all cotton terry. Stay away from paper towels, newspaper, and cheap Chinese microfiber.
Leo