Like any other filter, this one protects the front lens from water splashes. 

The same size. 

I have not looked for a specific travel case, but unless you are being really rough with it I would say a padded envelope or a generic small case would do it. It's pretty sturdy. I take it to to the office sometimes and just put it in one of the pockets of my laptop case. Also, for extra protection against scratches I never removed the clear plastic film that comes over the screen. Except for the red ""pull here"" tab, you can't tell it's there. 

No. You can't 

Yes, if you stack the 52-55mm and the 55-58mm adapter.