Who finds questions for and writes the New York Times crosswords? How do they determine what a good answer would be?

It may come as a surprise to find out that the New York Times crossword puzzles are not created by only one author. Usually each individual puzzle has one author, but there are a number of contributing writers who participate in creating the wide variety of puzzles published in the paper.

Different authors form their puzzles in different ways. Many writers try to incorporate at least one 15-letter word into the 15x15 square grid of the daily puzzle, but there are relatively few 15 letter words in common use. Due to the rules of crossword construction, which require diagonal symmetry of the black squares, if one 15 letter word is used, another will be needed to complete the construction. 

Many puzzles follow themes, and often construction is actually done in reverse. That is, the answer word or phrase is decided on prior to coming up with the question, because it is the answer that has to fit within more strict parameters in most cases (word length and letter placement, etc). This is not to say that coming up with the word clues is simple. A good clue is not too obscure or boring, vague, or overused. It sometimes takes crossword constructors months to create a puzzle they are satisfied with and also have it be accepted for publication. Often the process is quicker for skilled writers, but the creation of a crossword puzzle is a process of creativity and experimentation, and is often a puzzle in its own right.