Romeo is a very impulsive character. When we first meet Romeo, he expresses his sadness that his love doesn't love him back. He is talking of Rosaline. When he attends the Capulets' ball, however, he is immediately struck with love for Juliet:
Romeo. [To a Servingman] What lady is that, which doth
enrich the hand
Of yonder knight?
Servant. I know not, sir.
Romeo. O, she doth teach the torches to burn bright!
It seems she hangs upon the cheek of night
Like a rich jewel in an Ethiope's ear;
Beauty too rich for use, for earth too dear!
So shows a snowy dove trooping with crows,
As yonder lady o'er her fellows shows.
The measure done, I'll watch her place of stand,
And, touching hers, make blessed my rude hand.
Did my heart love till now? forswear it, sight!
For I ne'er saw true beauty till this night.
He dares to approach her, and they share an intimate moment. Juliet is also struck by him and expresses her love:
Juliet. Go ask his name: if he be married.
My grave is like to be my wedding bed.
Nurse. His name is Romeo, and a Montague;
The only son of your great enemy.
Juliet. My only love sprung from my only hate!
Too early seen unknown, and known too late!
Prodigious birth of love it is to me,
That I must love a loathed enemy.
Nurse. What's this? what's this?
Juliet. A rhyme I learn'd even now
Of one I danced withal.
This is what we might call "love at first sight." Do you think they are they really in love? The Nurse tells Romeo that Juliet is a Capulet and informs Juliet that Romeo is a Montague. Despite this knowledge, they still decide to pursue each other. I think Romeo is impulsive in his pursuit of Juliet. He goes to her window, despite knowing the danger. Do you think things would have turned out differently if they took more time to think about a potential relationship, or if they had approached it in a different way?
Romeo quickly switches his love from Rosaline to Juliet. Similarly, Juliet first expresses that she is too young to be married when her mother brings it up. A few days later, she marries Romeo.
One of the pivotal moments in the play is the sword fight that results in Mercutio's death. Romeo tries to stop the fight between Mercutio and Tybalt, but Tybalt stabs Mercutio. Romeo is understandably upset about his friend's death, but this leads to the rash action of fighting and slaying Tybalt.
Benvolio. O Romeo, Romeo, brave Mercutio's dead!
That gallant spirit hath aspired the clouds,
Which too untimely here did scorn the earth.
Romeo. This day's black fate on more days doth depend;
This but begins the woe, others must end.
Benvolio. Here comes the furious Tybalt back again.
Romeo. Alive, in triumph! and Mercutio slain!
Away to heaven, respective lenity,
And fire-eyed fury be my conduct now!
Now, Tybalt, take the villain back again,
That late thou gavest me; for Mercutio's soul
Is but a little way above our heads,
Staying for thine to keep him company:
Either thou, or I, or both, must go with him.
Tybalt. Thou, wretched boy, that didst consort him here,
Shalt with him hence.
Romeo. This shall determine that.
[They fight; TYBALT falls]
This action leads to Romeo's banishment, which sets in motion the fatal end of the play.
Hopefully these examples get you thinking about your paper! As for your introductory statement, as the opener of your paper, it should introduce what you will focus on. We know that you are writing about rashness and impulsiveness leading to deaths, but perhaps you want to focus on rashness in Romeo and Juliet falling in love too quickly, or rashness in the boys' fights that spark further conflict, or looking at all of Romeo's rash choices. Your introduction would then show us what the rest of the paper would look like.