"Is everything that happens caused by God?" YES! To say otherwise would be confessing Atheism or Deism.
"Does God cause people to do evil?" YES and NO! No in the sense that God sows evil in man's heart. Yes in the sense that He is the first and primary cause of all things and decrees all things. However, this does not mean that God sins or does evil because God is not a responsible being as man is; and God does all things for his glory while sinful man does things for his sinful end.
Is everything that happens a part of God's plan? YES!