I would say sentence 1 is correct. Sentence 2 is incorrect, we wouldn’t use the word “against” when talking about an exchange rate. Sentence 3, if you wish to be mainly mathematical and informative, can be changed to: “The dollar-yuan exchange rate is 7-1.” (The hyphen will replace the word “to” in this case.) Sentence 4 makes sense. Instead of saying the word “one”, you can also say “a dollar”. Sentence 5 is incorrect. You could change it to: “The Yuan is currently at an exchange rate of 7 yuan to one dollar.”

Personally I find them all ok. I suspect use of "rate" with a single number is probably commonplace. BTW: Using numbers, such as "1", rather than "one" was once considered lazy for small numbers.