Yes. 

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. 

with your TV and DVDThe same as the player, just ordinary household current. If you travel to Europe, you can take the charger with you, you just need to buy a plug adapter …There are different wall plugs in Europe. 

Yes you can! 

Yes, it can be automatically paired after being taken out from the case.