I don't think we can really know in any particular case, but there's a lot of evidence he actively tried to coin new words and phrases, so I could definitely believe that he was the first to use a number of those.
For instance, there are a huge number of words that, as far as we know, Shakespeare was either the first to use or the first to use in a particular sense:
http://thinkonmywords.com/additional/p161.html has a thorough breakdown of a good number of them. Now, a lot of them are variants on words already in use in other languages, but it's still pretty darn impressive, and a number of them are unusual portmanteaus he could well have coined.
And we do have a good number of written sources from around the time, including people who liked to use vernacular, so if a phrase were in common use before him, there's a decent chance we'd run into it.
So yeah, I'm pretty confident he's at the very least the person who introduced many of those phrases into widespread usage, and likely the very first user for a number of them.