I agree with 3 and 4. Fate is a huge part of the story. Shakespeare is suggesting that these events are fated, and there are several allusions to the star-crossed lovers line in the prologue. This is one of them. Romeo is indicating that he is at the mercy of fate. This idea is a large part of the motivation for some of his actions later in the story.

"O, I am fortune's fool" is unmistakenly a reiteration of the motif of Fate that is introduced in the very Prologue to Shakespeare's Romeo and Juliet that mentions "star-crossed lovers" who are "from the fatal loins."  Impetuous and impulsive, Romeo does not feel that his actions are performed entirely of his free will. In Act III, he blames some of his action upon fate. Later in the play, in Act V, Scene 1, he learns about Juliet from his servant and exclaims, "Then, I defy you, stars!"  Here, Romeo decides he will take action against fate.

I don't think that he means "fortune" in the sense of being "fortunate."  I think that he means "fortune" in the sense of "luck" or "fate."  I think that what he is saying is that fate is basically toying with him. Everything that has happened to him seems (to him) to simply be bad luck.

This is important to the outcome of the story simply because it gives us one way to interpret what happens.  It is implying that all that happens to Romeo and Juliet happens simply because they are "star-crossed."

Romeo has just killed Tybalt. He could consider himself fortunate to be alive, but a fool for killing his kinsman of one hour.

Romeo has just married Juliet, only an hour before he kills Tybalt.

He could consider himself fortunate to be married to Juliet and and a fool for killing her cousin Tybalt.

Either way, Romeo is in trouble. He is fortunate to be married to Juliet. He is fortunate to be alive, but he is a fool for killing Tybalt.

Ironically, this statement is possibly foreshadowing the end of the story. Romeo is fortunate to have loved Juliet, but he is a fool in the end because he kills himself, unaware that Juliet is only sleeping.