1) This is old "folklore" style advice, the kind of thing that is sometimes called an "old wives' tale." The general idea is that your return to health will be quicker if you eat well while you have a cold (to keep up your strength) but refrain from eating if you have a fever, presumably because the fever will interfere with proper digestion, or you might even vomit. Many people have trouble keeping the expression straight, though, and say "starve a cold, feed a fever." It's also sometimes used humorously, e.g. someone with a very mild cold stuffing his face with a burger, fries, and a chocolate shake who says, "Well, you know what they say---feed a cold..."
2) A boarding school is a school where the students live on campus in dormitories. Hogwarts is a boarding school. The "board" part comes from "room and board" (or sometimes "bed and board"), an somewhat old-fashioned way of referring to where you sleep and what you eat. (Think of dishes of food laid on the "board," or table.) "Boarding houses," where residents paid a monthly rent for a bedroom that also included one or more daily meals eaten at a common table with the other residents, were once quite common.
"Au pairs will receive travel expenses as well as room and board and a small monthly stipend."
3) Two times a month is fine, but twice is a little more usual. Two times a day, or even week, would sound awkward (but not incorrect) to me.
4) I believe that "the kiss of life" is an old-fashioned expression for mouth-to-mouth resuscitation, or artificial respiration. I've never heard it, but I think I've seen it used in (British?) novels. I can imagine a lot of people, in the US at any rate, would look blankly at you if you started talking about the kiss of life.
[not a teacher]