What did the Isrealites do wrong?
After God protected them from plagues, delivered them from slavery in Egypt, gave them the gold of their captors, miraculously parted the Red Sea for them to cross, provided daily food and water, and even clothes and shoes that did not wear out...you would think they might be at least a little grateful. But no, they whined and complained about almost everything, including Moses' leadership. Then, in a final insult to the Almighty God, as soon as Moses was out of sight upon the mountain, they had Aaron fashion a new god, a golden calf, out of the very gold God had provided them. No wonder God was angry! (Unfortunately, it seems that even in modern-day America, blessed as we have been, our behavior is much the same).
How can a loving God be angry?
I think His anger stemmed from His great love for these people He himself had created, as well as His complete holiness. Nobody can make me as angry as the people I care most deeply about. If I read in the newspaper of some evil deed done by a stranger, I shake my head in wonder or disgust, then go about my business. But if a dear friend or family member of mine is involved, I may fume for days (and sometimes nights too). There is no end to my distress until something is resolved.
Is God's sentence to destroy Israel and raise up a new nation through Moses justified?
Yes, entirely. Moses was not without fault, but he obeyed God when no one else would. He had a yearning to know God on a personal level. And since he was a descendent of Abraham, he could father a new nation in fulfillment of God's promise that Abraham's seed would produce a great nation.