I throw mine in the washing machine with the rest of the laundry and they get tumble-dried. Is there any reason you can't do this? Is there something I don't understand?
It's important to wash them. The sweatband is there to absorb sweat and if you never wash the cap, the salt and magic stuff in your sweat will rot the material. (Magic stuff: enzymes? Acids?)
Eventually a cap will wear out. Cotton doesn't last forever and, providing you treat the outgoing cap with respect, you may retire it.
No special ceremony is prescribed and every driver has their own.
It is important that you have been breaking in a new hat for the previous month or two. Every new hat is a commitment and a responsibility. A new hat, though, needs breaking in gently. A day or two at a time, over the several weeks, will let the hat relax into its new role.
I suppose if you want to be extra fussy with your hats you could hand-wash them in one of those detergents intended for wool, then rinse them very thoroughly to get rid of a) the luverley perfume, and b) excess detergent that will be of no benefit, might irritate your skin, and perhaps also degrade the material.