1. The last book is divided into two movies because of the financial success of the last two Harry Potter movies and last two Twilight movies. Both final books in these series were made into two movies, so Lionsgate followed suit (basically, it's for the money.)
2. Peeta drew the picture of Rue because he wanted to make the Gamemakers feel guilty about Rue's death.
3. District 12 was blown up by the Capitol. Due to Katniss being the symbol of rebellion, President Snow sent Capitol hovercrafts to the district to blow it up.
4. At the beginning of Catching Fire, Snow's plan was to have Katniss convince the citizens of Panem that her attempt to eat the berries at the end of the Hunger Games was not an act of rebellion, but an act of love. However, this failed, and so Snow set up the Quarter Quell so he could get Katniss to re-enter the arena and, hopefully, die.
5. Are you asking how Haymitch got involved in the rebellion? Basically, he was approached by Heavensbee and other rebels and asked for his cooperation. He kept Peeta and Katniss out of the loop until the force field in the arena was blown up.
6. The three-finger symbol is a sign of respect/farewell. The people were killed when they did this because they were showing their respect for Katniss, the Mockingjay, the hero, the rebel. Thus, respecting the rebel is an act of rebellion and defiance.