I have the Tilley TH5 (natural) for hot days when a lighter shade is best and a Tilley TWC5 in waxed cotton (British Tan). I like them both but wear the
TWC5 Outback Hat more often. Both are very comfortable and work well providing shade. Not much rain here in SOCAL so I can't speak to one of these in the rain, but I s'pose the waxed cotton TWC5 would do well.
I'm thinking not crushable. If I were to pack it, I'd pack the inside with socks and T-shirts and take my chances. It would probably be fine.