Yes both of those are possible. I prefer 'go out for dinner'. For me, 'go out to dinner' seems just a little more formal, but they are both fine.
For the second one, it is possible to say 'Amazon's', but I would say that we don't really use this with proper nouns too much in written English. This is because it looks too much like a possessive: 'The Amazon's amazing wildlife.' 'The Amazon's many tribes'. I would definitely prefer 'The Amazon IS bigger than the Nile'. In spoken English, however, this construction is a lot more common and I use it all the time.