A letter of inquiry is a kind of letter that is written in order to request information and/or ascertain the authenticity of an information. Thus, the major purpose of writing an inquiry letter is to ask for relevant information about a particular matter. Inquiry letter deals with various matter such as vacancies, funding,grants, scholarships, projects, etc.
I'm not sure about number 1. Maybe: It was the happiest event. 2. The judge cautiously handed down her decision. 3. Mr. Majors was pressed hard to find an answer.