1) "You are short a P.E. credit to graduate" is just a very informal way of saying "You are one P.E. credit short of graduating." "You are short a P.E. credit to graduate" sounds natural to me (I live in Arkansas, in the U.S.), but it might sound odd to native English-speakers in other places. 2) "This is short on salt" works perfectly, but where I live we'd actually say "This needs more salt" or "There's not enough salt in this." I would say "short on" in other sentences, though, like "I'm short on sleep," or "We're short on time," or "He's short on money." What's the context for the sentence "This is short three dollars"? I'm imagining this situation: A customer hands a cashier some money to pay for groceries, but he miscounted the money he just gave. The cashier might say "You're three dollars short," but he wouldn't say "This is short three dollars." At least not where I live. 3) This one's complicated, and it's also probably different in different places. If I were person B, I wouldn't say it like that at all -- I'd say "I never saw them" or "I never got them." If you absolutely have to use the sentence "They never came ______ my office", the only one of those three words that works is "through." Think of it this way: The records were going from person A's office through person B's office and then on to somewhere else. I would only say "They never came in my office" if I were talking about people, not if I were talking about patient records. I would never say "They never came over my office." Sometimes, though, you'll hear someone say "They never came across my desk." 4) I'm not sure on this one, but "records officer" sounds like the person who's in charge of maintaining records, and the word "deputy" means that there's another records officer who's been working there longer and who oversees the deputy.

Lee and Gertie are right: "the sound made by my mom when she cooks" and "the sound of my mom cooking" both work, but "the sound of my mom cooking" sounds more natural. Where I live (Arkansas, in the U.S.), "the sound of my mom cooking" is a bit formal. I would say it like that if I were writing an essay, but if I'm talking with friends I would say "the sounds my mom makes when she's cooking."

Hm... I don't know if it's different in other English-speaking countries, but in the U.S. (at least, in my part of the U.S.) I never hear "on to" being used like that. Occasionally I will say or hear the phrase "move on to [noun]" or "go on to [noun]" in sentences like "Let's move on to today's lesson," or "I'll go ahead and go on to the next question," but that's about it. It's pretty rare, at least where I live.

1 "Alive" is correct. Grammatically, I think "live" is also correct and it can mean the same thing, but nobody ever says it like that. 2 This one's tough. I wouldn't use either of those words. I would say something like "to end racial discrimination" or "to stop racial discrimination." In my opinion, the word "dissolve" is more for an organization ("He dissolved the company"), and the word "settle" is more appropriate when you're talking about an argument ("That settled the issue"). But see if someone else has a different opinion. 3 The correct word is "oncoming." The words "oncoming" and "ongoing" look like opposites, but they are actually completely different. "Oncoming" means "coming this way", and is usually used when talking about traffic (like in this case). "Ongoing" means "currently happening." It comes from the phrase "going on." So in a sentence like "they had been fighting the war for a long time," you could describe the war as "an ongoing war."