>How many wearings does it go through before a wash?
No set number. I wash it once the neck and/or cuffs start showing dirt. I keep 3 of the same white dress shirt.

>What's your method of washing it? Dry Cleaning?
Wash it with white clothes, and hang until almost dry and iron while a bit damp.

Perhaps I can just handwash it?
I hand wash from time to time. Works well.