"We do not read the letter" sounds very strange in english. We did not read the letter would sound more natural. Because it's definitive, it sounds as for example, if you received a letter after someones death. We do not read THE letter (EVER) makes sense, but it wouldn't be a very commonly used sentence. We did not read the letter however, sounds less important. We didn't read it, but would/could later.
As a general comment, you have a few translations into english which although not wrong do not sit right on the ear. I am reminded of a Churchill quote, "Up with which I will not put." Which he explained, while perfectly correct grammatically no englishman would ever express in this way.
Out of curiosity, in English, this sentence ("we are not reading the letter") would be interpreted generally as "we are not going to read the letter" (i.e. it will never happen) rather than "we are not currently reading the letter" (i.e. it isn't happening at the moment).
Is this true in Swedish as well or no?