1) The opening quote is considered to be part of the word, so the underline symbol is first, followed by the space, open quote and word, space termination sign. Otherwise, the open quote would be separated from the word by a space.
2) That would be either way. I would use contractions for the underlined words as well as everything else. This is about word USAGE, not spelling. This is not about how to spell the word WHOSE, for instance, but whether to use WHOSE or WHO'S. Likewise with YOU'RE and YOUR and IT'S AND ITS. I think using uncontracted braille for these words would be a bit of overkill, but it would not be incorrect if you chose to do so.