Yes, you can attach the phone lens to the main phone camera of iPhone 13. 

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

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. 

Yes there is a case/stuff sack and as far as accessories there are two batteries, charger, and 32GB MicroSD card. Pretty sure there was a lens cloth as well. 

Yes it does...I have a Spigen case on my iPhone 13 Pro Max and fits no problem at all and that clearly is bigger than a standard 13 Pro. Regarding the Gimbal, it's a great bit of kit and works brilliantly!!