How to know if there is a checkmate in n number of moves?

How to know if I should look for checkmate in n number of moves? (Mate in 1 is the easiest to find, then 2 is a bit harder. So I guess n>2 is a reasonable condition here.) In other words, what are the signs that I can checkmate in some number of moves?

And once I have a (good) hunch that there could be a checkmate-able line waiting, how to find out the number of moves? How to find that number quickly?

It's just experience. If you do puzzle Rush, you'll see certain mate themes come up repeatedly.   You may notice those situations coming up in your games.

There is really no secret principle. You have to calculate all variations. To develop the intuition that tells you that there might be a forced mate, you have to improve your pattern recognition.

Quite often, it happens when one player is severely up in material or has pieces directly attacking the opposing king.

Calculation is still important as your opponent will try and defend the position to his best ability.