"My trying times" is a bit old-fashioned, but is still used a good deal, especially in writing. A related noun is "trial" -- "Climbing stairs can be a trial for older people." Also common: referring to troubles as "trials and tribulations"

They are similar if they refer to a person not being present, but hiatus is used more broadly than absence. You can say, "the author took a hiatus from writing" but not "the author took an absence from writing."

No, they are not interchangeable. #1 asks if the brother is at the other person's home, which #2 asks if the brother is with the other person. The brother could be at the other person's home, even if the other person is not with the brother. And, re #3, the brother could be with the other person, but somewhere other than the other persons' home. #2 seems to imply that the other person kidnapped the brother. That is the only situation in which an English speaker would refer to someone "having" another person.

#2 should read, "Having taken my advice, you will be successful." "would be" implies something that did not occur, but would have, had the person acted differently. eg: If I had gone to law school, I would have become rich." Versus, "having gone to law school, I will become rich."

Committment to education does not mean the same thing as committment to children. "Committment to children" is more broad. An example from current events: If a child tells his teacher that they are transgender, but does not want her parents to know, should the teacher tell the parents? If a school is "committed to children," the most likely answer is "no." If they are committed to education, the answer might be "It depends on whether that will affect the student's ablilty to learn."

It is "four by 100." Similarly, if you want to buy a common size of lumber, you would go to the lumber yard and ask for a "two by four" (2x4 inches)

The second usually refers to past action. Eg: "I am writing a novel" = I am currently writing a novel "I have been writing a novel" = I have been writing a novel for some time. But it does not necessarily mean that you are still doing it. "I have been writing a novel, but recently I have been so busy at work that I have not had time to write." It is a little odd to say "I have been setting out to become a doctor," but it is not wrong, because it could make sense in context: eg, you have taken biology classes, have studied for the Medical College Admissions Test, etc. In other words, you have taken several steps required to become a doctor.