In (3) you have said 'letter' instead of 'letters' - I think just a typo. Apart from that all are grammatically correct - (1) means that you are almost certainly the person to write the letters (2) and (3) suggest that someone else might be writing them for you.

Correction: 3. There are some letters to be written. I think #1 and 3 are grammatically correct. #1 is more common. #2 actually does not exist. #3 sounds a little stiff. #2 can be paraphrased: "I have some letters that need to be written." I have some letters to write. = There are some letters to be written. = I don't have those letters, but I have an obligation to write the letters.