Use of English articles for proper nouns/names

Hi. Let us pretend there is a restaurant named "The John Doe's," with the word/article "The" in front of it. I think, eventhough I could be mistaken, that the names such as this has the word/article in small letters if written in sentences (as part of sentences) like this:

She met him at the John Doe's yesterday.

I also think you could write the first letter of the word/article in the sentence above in a capital letter like this (I think this is correct, too):

She met him at The John Doe's yesterday.

It also seems that I think we should put the letter "t" in a capital letter for the word/article "The" in the name "The John Doe's" if we are to put that name under the picture of that restaurant that was to be put on a cork board, to be used as a display.

Also could you answer these three questions?

1. Could we write "She met him at "She met him at John Doe's Restaurant" if the name "John Doe's Restaurant" is in a sentence? (Note that the name is supposed to be "John Doe's Restaurant" with the letters "J" and "R" in capital letters.)

2. Could we write "She met him at a John Doe's restraurant" if the restaurant "John Doe's" (without the article "The", as in the previous case, top above) is a chain restaurant.

3. If the name of the famous museum, "The J. Paul Getty Museum", is to be included in a sentence, would you write "She met him at The J. Paul Getty Museum" or "She met him at the J. Paul Getty Museum"?