As far as I know in the muslim religion Jihad is a part of the religion?
As much as holy war is part of Christianity.
"Onward, Christian soldiers, marching as to war,
With the cross of Jesus going on before.
Christ, the royal Master, leads against the foe;
Forward into battle see His banners go!"
What jihad really means in various branches and interpretations of Islam would be a very long trek to get into, so at this point I'll just assure you that, in general and by and large, it doesn't have a great deal to do with what you have in mind (i.e., attacking infidels and converting them by force).
I also think the death punishment is a part of the koran ?
The Bible too.
Edit: Oh, and... I agree with POLYGON that religion isn't the issue in the current Lebanese crisis. It's about politics and group identity, which overlap and cross in a number of complex ways. The closest we've come to pure sectarian conflict this time around was in Tripoli, and there the Sunnis didn't attack the Alawites because they think the Alawites worship in a wrong way, follow wrong dietary rules, or celebrate the wrong festivals; they attacked them because over there the Sunnis are pro-government and they believed the Alawites, being Alawites like the Syrian president, are pro-opposition. Politics and group identity, not religion.
Come to think of it, I don't think I've ever once heard a Lebanese use a religious argument when explaining exactly why group X should be run out of the country; OTOH various class, race, and clan arguments are common, and most common is a very long-winded discussion of the wrongs suffered at the hands of group X. This applies even to overtly religious groups like Hezbollah.
Edit: scratch that, I remember a party with some Christians, most of the Aounists, where they eventually started drinking toasts to Slobodan Milosevic for having killed so many Muslims, and Adolf Hitler for having killed so many Jews. I still think even that was more about group identity (racism) than religion, though.