The Montagues plan to enter the Capulets' house by disguising themselves and blending in with the guests attending the Capulet's masquerade ball. This allows them to avoid detection and gain access to the enemy territory without being recognized.
Tybalt has vowed revenge upon Romeo and has sent a challenge to Romeo's father's house.
"House" here must mean "family"; Romeo's family is the Montagues and Juliet's the Capulets.
Romeo and Juliet opens with the 'Chorus' telling about Verona and the feud between the Capulets and Montagues. Then, two servants of each house begin to argue and eventually pull their swords. :)
Mercutio is expressing his anger at the Montagues and Capulets for having the feud which he is blaming for his death. Of course, Mercutio is neither a Montague nor a Capulet, and therefore the feud was not the cause of his death. The real cause of Mercutio's death was his belligerent nature. He sought out a quarrel with Tybalt and has reaped the reward of it. But he doesn't want to face that truth, so he keeps on cursing the feuding families in order to avoid facing it.
Romeo and his fellow gate-crashers are wearing masks which makes them hard to identify. This is ok, as it was common for people to come to parties in disguise like this. Capulet remembers doing it when he was younger. However, Capulet is in an expansive mood, and even after he knows that Romeo, a Montague, has crashed the party, he does not object to him staying and having a good time like the other guests. Of course, Mercutio had an invitation, so he didn't have to fake his way in.
Tybalt. He sent a letter to Montague's house challenging Romeo to a duel.
Because Sampson Bite His Thumb At Them. Actually- Sampson and Gregory are servants of the Montagues; and they fight with the capulet's men because the Montagues and capulets have an "ancient grudge" against each other and they want to cause trouble to the capulet men upon acknoledgement of them.
At the Capulets' house
He is calling a plague down on both the Montagues and the Capulets because their feud has led to his death
It was never stated, though he's a kinsman to Prince Escalus. Neither was Paris's last name, for the record. Benvolio's last name was never specified, though he was from the house of Montagues. He's a cousin of Romeo's, so it's possible they had different last names.
Tybalt sends a letter round to the Montagues' house to challenge Romeo.
If I remember correctly it is the Capulet party, and is the only party in the whole play. Of course, by me saying that, the reason Romeo thinks it's a bad idea is because it is the Capulet's house.