I think I'd find it a little too cute if I got the synopsis in 1st person--I like it when some of the author's/narrator's personality creeps into the synopsis, but I don't want a personal letter from the character, as it were. I need the synopsis to be a quick read that reassures me the end of the book holds up. I'd err on the side of fewer flourishes rather than more.
I agree. If I read a synopsis in the form: "I did this, and then I did that, and by the end, I got the girl" (worse, if it's in present tense), I'd probably think "this guy doesn't know what he's doing.
I mean, imagine you have to tell an agent, face to face, about your story and plot -- say, in an elevator. That's pretty much what a synopsis is. It would be really weird if you start by: "So I jump off the cliff and break both my legs..." It suddenly sounds like a real, biographical story... it may work, but I probably wouldn't chance it personally.
And yeah, you may find an agent who entertains it, but chances are they won't, and why risk more rejections?