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Usually, each maths textbook defines the syntax used within at the start and then ‘rewrites’ all proofs in that particular syntax.

Since syntax – as opposed to the mathematical facts expressed thereby – is usually considered something not worth learning by heart, I doubt that you will find a book that collects proofs in their syntax of original publication and then explains that syntax.

I wrote a little "math syntax" to "english words" dictionary which might help you: http://mcgillweb.alwaysdata.net/notation_appendix.pdf

In particular the section on set theory is important (element of, subset, etc..) and the two qualifiers: for all, and there exists. Here is a 6 page primer on set notation, which contains a simple proof http://mcgillweb.alwaysdata.net/set_notation.pdf

Yes!

Grab a copy of "Proofs from the Book". It's tremendous.

Proofs from the Book is awesome. Another book in a similar vein, but requiring less of a mathematical background and targeted more at the curious layman is "Journey through Genius". Journey through Genius also spends more time giving general background about the players involved and telling the story leading up to the proof.