The diameter of the biggest part, namely the camera head, is 23mm/0.9in. 

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, it charges them quickly and then the light for that slot turns green. To me, that's an indication that it's going to lose power because the battery isn't hot the next day. (No, they don't need to be charged overnight, they're fast.) 

I don't know the exact size but it even fits 12-13 year old kids.