Can it be washed? Either hand washed or in a washing machine?

Yes we throw it in the washer (not the cooling part)

Can you fold it for packing?

It mashes flat- I don't think I would fold it. The crown flattens, and you could put it on top in a suitcase- even if you thought it was full. Very thin.

Would this fit my husband who requires a hat size of 7 3/4?

yes- but I would skip this one- it doesn't work any differently than any other hat and it stains very easily when you sweat

Is there a barrier between the water added to the cap and your hair? Or does your hair get wet?

There's no barrier. Your hair will get a bit damp on top but most of the water will stay where you poured it. Think of it as using a sponge or damp towel on top of your head.And in fact it DOES make a difference, I use the hat in an Arizona summer and as far as I'm concerned it's already paid for itself several times over.