God destroyed the world with a flood, destroyed all the people in Sodom and Gomorrah, gruesome sacrifices, laws for stoning people, Jericho, requesting the sacrifice of Abraham's son, the sacrifice of God's own son Jesus'. Jesus seems so different to me because instead of making life or taking life he gave his own, he preached forgiveness and grace and seemed to defy the laws made in Leviticus. I'm a christian but I am having a hard time understanding this, especially if one believes the Trinity doctrine.
If you look at the role of Jesus especially at the very end of time and the beginning of eternity (eternity for us, that is to say) he does not seem that different after all. When Jesus returns, he will not return as the contemporary hippie Jesus. That image is a product of modern humanistic culture and not the Jesus of scriptures. He will return as the King of kings, terrifying in his righteous wrath. That will be a day of devastation for anybody who does not want anything to do with him.
One striking difference between Noah and the flood vs Abraham and Sodom/Gomorrah is that Abraham was the one who thought that perhaps there was still hope for the two cities. Noah did not. Abraham pleaded with God and God agreed upon a minimum number of righteous people who, if present, would be grounds for not carrying out the punishment. A similar incident happened during the early exodus, when God, appalled at the golden calf the Israelites had made, offered Moses to destroy them all and make him the new god's people. Moses turned the offer down.
It is almost as if God tries to provoke a reaction. If Moses' reaction had been "yes, let's do away with them wrongdoers" God most certainly would have carried out his threat, as his righteousness demands. God never bluffs. At the same time He would have been disappointed in the hard-heartedness of Moses.
I believe this type of thinking can be applied to the mosaic law. It gives a human being an apparent right to stone another, but it does not take away the possibility to show compassion. The problem is that the israelites rarely did that. It tells more about man than it tells about God. The tendency to point fingers at each other and not see our own error is a product of the god-syndrome mankind fell victim of in the garden of Eden. That is something Jesus pointed out, and that is just what he did; he did not introduce a new hippie religion or change, defy, or undo the law, but pointed out how the law should have been applied in the first place. He explained what the law really is and is supposed to be; a dilemma.
I hope this answer helps and manages to touch all the different issues in your question.
There is not difference. Jesus, like all the other OT prophets, preached that judgement was on its way. Though he did offer forgiveness, it came with an "or else."
Matthew 24:37-39 King James Version (KJV)
2 Thessalonians 1:8-9 King James Version (KJV)
Many other judgement verses in the NT could be cited, such as most of the book of Revelation.
I speak only for myself here.
If you read through the OT quickly, you will "see the forest for the trees", that is, you will get an overarching picture of who God is and what His purpose for us is. This will often get lost if you focus only on the details of the OT.
Many scholars have discussed the way God interacted with an ancient Near Eastern people, and there are good sources* to read about that. There are some who don't believe in the genocides, the flood, etc.
Even if these narratives are true, the OT teaches us about the Holiness of God, His lovingkindness in the face of our repeated faithlessness, His allowing us free will and to reap the consequences of our sins, and our need for an intercessor - His son. At the appropriate moment in History, He sent us the intercessor, that was foretold in all of the OT. In essence, I don't think we could understand the magnitude of our separation from God to truly appreciate the sacrifice of His son without the OT.
Any attribute of Jesus can be found in the Father. Also, to underscore God's place, the Greatest Commandments are “'Love the Lord your God with all your heart and with all your soul and with all your mind.' This is the first and greatest commandment. And the second is like it: 'Love your neighbor as yourself.' All the Law and the Prophets hang on these two commandments.” Jesus does not, ever, contradict any part or action of His father.
*Inspiration and Incarnation: Evangelicals and the Problem of the Old Testament, by Peter Enns