The short answer is "To teach Jacob an important lesson."
This all takes place, of course, in Genesis 32.
First, to your first question: Jacob wasn't "good" by any means. If you look at Jacob before this, he has stolen the blessing that was Esau's birthright. (Genesis 27). He had several children with women he didn't love, and was certainly no great father to them. (Genesis 30-31).
By the time he wrestled with this angel (or with God depending on the translation - I'm going to stick with God) in chapter 32 starting in verse 24, he'd pretty much shown that he was not a "good" man at all.
Yet, God chose him to the the father of the nation of Israel.
To the second question: was God trying to kill Jacob? In the wrestling match, Jacob wrestles with God until daybreak (verse 24). Then God touches the hollow of Jacob's thigh and dislocates it, demonstrating that He could have easily defeated Jacob at any time. This was a lesson in humility - showing Jacob that compared to God, he was nothing.
That also answers the third question. Jacob was injured - he was disabled by a mere touch, to show that God is powerful and compared to God, he is nothing. Whether he was "seriously injured" depends on whether or not you consider a dislocated hip "seriously injured".
Yet, knowing full well that he was nothing, Jacob did something brave and remarkable. He refused to give up until God gave him a blessing. He continued to fight, but now it was for God, not against God.
I believe that the reason for the encounter was to ensure that Jacob, who became Israel afterward, had the correct motivation, heart, and attitude toward God. it was a lesson in humility, and Jacob responded by a show of faith and longing for God.