To what extent is Romeo's tragic flaw responsible for his downfall and sudden death? Please provide a quote from when Romeo kills Paris, as well as one from when he hugs Tybalt, and another about how Friar Lawrence is responsible for Romeo's death.

Romeo's tragic flaw of impetuousness is responsible for his downfall because his rash behavior leads him to make poor judgments, decisions that bring about his death.
When he first sees Juliet, Romeo exercises no caution; instead, in the Capulet home, he boldly approaches Juliet, asking her for a kiss: "My lips, two blushing pilgrims, ready stand / To smooth that rough touch with a tender kiss" (1.5.91-92 ). His actions incense Tybalt, Juliet's cousin.
Having won Juliet's heart under her balcony, Romeo rushes to Friar Lawrence to ask the priest to perform the marriage ceremony for them. Amazed at Romeo's change from being lovesick over Rosaline to being rashly passionate about his enemies' child, Juliet, Friar Lawrence is reluctant to marry them, but he does not want them to live in sin. He tells the young lovers, "...you shall not stay alone / Till Holy Church incorporate two in one" (2.6.36-37). After they are married, the impulsive Romeo quickly changes his feelings toward Tybalt, since Tybalt is part of the Capulet family to which Romeo now belongs. This new relationship is the reason that Romeo impulsively professes his love for Tybalt amid a heated situation. For, in his effort to diffuse the conflict between Mercutio and Tybalt in the public place in Verona, Romeo tells Tybalt

...the reason that I have to love thee/Doth much excuse the appertaining rage [the anger with which he otherwise would have responded]/ to such a greeting. (3.1.34-35)

The enraged Tybalt, who knows nothing of the marriage, argues that what Romeo says does not excuse the injuries that Romeo has committed against him. Still, Romeo tries to convince Tybalt that he "doth love better than thou canst devise"(3.1.40). Unfortunately, Romeo's rash efforts to intervene and diffuse the situation fail and his friend Mercutio is slain by the angered Tybalt.
Ironically, also, it is the priest's rash judgment that leads to Juliet's suicide because Friar Lawrence runs out of the catacombs without forcing Juliet to accompany him when she insists upon staying. Indirectly, too, he has some responsibility for Romeo's death since Romeo would not have killed himself if Juliet were not seemingly dead from the potion given to her by the priest that causes Balthasar to report her death. In addition, Friar Lawrence should have made more efforts to ensure that Romeo would know about Juliet's having taken his potion to feign her death.
Without knowing that Juliet is really alive, because he has believed Balthasar's report that Juliet's "...body sleeps in Capels' monument [tomb] / And her immortal part with angels lives..." (5.1.19), Romeo then buys poison and rushes to the Capulets' tomb. It is there that Romeo, unfortunately, encounters Paris who says that he will "apprehend [Romeo] as a felon" (5.1.69). They fight and Romeo slays Paris.