Yes, it is a noun.
English parts of speech are extremely flexible, and almost any word can sometimes be used as another part of speech. 'A high' is, as you say, the highest point. The day-time high is the highest temperature that it is likely to reach that day.
And yes, this is Fahrenheit. The USA is, as far as I know, the only country which is still stubbornly clinging on to Fahrenheit. An American 45 degrees is just over 7 degrees Celsius.