She got invited to Capulet's party, which made Romeo want to crash it.
Yes. He says "If I profane with my unworthiest hand . . ."
At the opening of Romeo and Juliet, Benvolio teases Romeo about being in love with Rosaline. Romeo is having no success with Rosaline (and in fact he goes to the Capulet's ball because he believes she will be there). We do not know if Rosaline is Romeo's first love - for all we know he may have had many girlfriends before her. But she never appears in the play, and as soon as Romeo sees Juliet he forgets all about her. I thought that he went becasue his friend made him go!
Rosaline Capulet is only fifteen years old. That one makes her only one year younger than Romeo and one year older than her favorite cousin, Juliet.
To Capulet's party. Benvolio wants to go to get Romeo's mind off Rosaline; Romeo wants to go because she is one of the guests. They plan to sneak in under the wing of the invitation to Mercutio.
Benvolio tells Romeo that Rosaline will be there, and Romeo can look at her all he wants when he is there, and he can wallow in his misery at being rejected by Rosaline.
The phrase "I conjure thee by Rosaline's bright eyes" suggests that the speaker is using Rosaline's beauty or presence as a way to invoke or compel someone to do something. It implies a sense of urgency or persuasion, drawing on the power of Rosaline's allure to influence the person being addressed.
They observed lots of customs. We see that Mrs. Capulet, being wealthy, had a wet nurse for Juliet, which nurse was kept on as a companion and a servant. We see that young, marriageable girls were kept mostly at home. It was usual for women to play hard-to-get with suitors, like Rosaline does.
Her name was Rosaline. Rosaline wasn't interested in him. (No, she was not a nun. The idea of someone expressing romantic interest in a nun would have been even more offensive back then than it is now, unless she was from that other kind of "nunnery" videlicet a brothel.) Romeo didn't so much love Rosaline as fall in love with the idea of being in love with her. That meant he could mope and moan and talk in incessant oxymorons, and never have to make any hard choices or sacrifices for the woman he loves.
Which woman? Rosaline? Juliet? The Nurse? He is apparently upset about Rosaline not being interested in him, as was the custom. He is very upset about not being able to see Juliet because he is banished from Verona.
Not at all. He is playing at being a man in love, just as she is playing at being cold and aloof. These were conventional attitudes. Romeo doesn't really care about Rosaline. When he meets Juliet, it is like a slap of cold water in his face.
He tried to attract Rosaline with fancy poems, songs, and big bunches of flowers. I don't know where that appears in the text. In the play he is trying to attract her by moping around being depressed.
Tybalt is Juliet's first cousin, being the son of Mrs. Capulet's brother. By marrying Juliet, Tybalt then also becomes Romeo's cousin by marriage.