Does the brim of the hat flap around in the wind or stay in place if, for instance, you are driving around on a fast lawn mower in a breeze?

I play golf often and the hat flap does not - unless the wind gets 25 mph and then it's still not too bad. I had the same trouble with other wide brim hats blowing up if the wind got too high. I rarely have any "flapping" issues with this hat.

How's the sun protection with the mesh on the side?

Not a problem. I live in Arizona, and the mesh is a must in the heat. A lot of upside and no downside.

Is there any size adjustment?

My hat size is 7 5/8 - would this fit my head comfortably?

How do I get rid of this crease from the hat being folded in the package?

Put it someplace where the hat is in it’s hat shape. Also helps to wear it frequently. After two weeks being “flat” instead of folded, the hat will start to prefer being flat to being folded.