What are examples of the things that happen to people that are "fated/predestined" which God is responsible for?
I'm glad you asked.
There are certain things to which man is forced and compelled, such as
sleep, death, sickness, decline of power, injuries and misfortunes; these are not subject to the will of man, and he is not responsible for them, for he is compelled to endure them. Man is compelled to endure them because he has no choice, and since they are not chosen that means they are fated (e.g., it was my fate that I got hit by a car while riding my bicycle, I did not choose that.)
Question.—Is man a free agent in all his actions, or is he compelled and constrained?
Answer.—This question is one of the most important and abstruse of divine problems. If God wills, another day, at the beginning of dinner, we will undertake the explanation of this subject in detail; now we will explain it briefly, in a few words, as follows. Some things are subject to the free will of man, such as justice, equity, tyranny and injustice, in other words, good and evil actions; it is evident and clear that these actions are, for the most part, left to the will of man. But there are certain things to which man is forced and compelled, such as sleep, death, sickness, decline of power, injuries and misfortunes; these are not subject to the will of man, and he is not responsible for them, for he is compelled to endure them. But in the choice of good and bad actions he is free, and he commits them according to his own will.
For example, if he wishes, he can pass his time in praising God, or he can be occupied with other thoughts. He can be an enkindled light through the fire of the love of God, and a philanthropist loving the world, or he can be a hater of mankind, and engrossed with material things. He can be just or cruel. These actions and these deeds are subject to the control of the will of man himself; consequently, he is responsible for them.
Some Answered Questions, p. 248
You can read the whole chapter on free will on this link;
70: FREE WILL