How do letters to my dr work?

so i know that you can write letters for characters in your dr and that they will respond you with a sign that you can chose but i have some question.

1)what is the avarage time for characters from your dr to actually respond? Does it take days? Weeks? Or even a month?

2)what if they don't respond? I always wondered this because how about i write them a letter and they decide that they don't want to respond?

3) is it garanteed that they'll receive the letter?