Who is the God of the OT and Who is the God of the NT?

Jesus is in the OT and the NT. It is the same God. Get a study Bible and look up the places in the OT that were prophecies of the coming Messiah and then look up the places in the NT where those prophecies were fullfilled. It will really open up your eyes to the fact that the whole Bible from beginning to end is about Jesus. As to your description of God in (a. ) I see someone who delivered the Israelites out of over and over. I see a God who gave them victories in war when people would have taken them as slaves. I see a God who gave them manna to eat in the desert. I see a different God. I see a good and merciful God in all of the Bible. Jesus is God.