Just something I'd like to mention, that no one else seems to have.

The letter is written in first person, correct? (Using the word "I", etc.)

In the Japanese version of the game, the "I" that is used is one that is specifically and only used by young boys. Vivi is the only young boy that has enough relevance to the story and the letters contents to be the writer. And as mention at the beginning of this thread, it was officially stated by Square as well. I don't see why it is so hard to understand that Vivi did indeed 'stop'. The letter was meant to be a sad and poetic way of telling us that. Does everyone really need him to say, "Good God, I'm going to croak soon!" in order to believe this?

As for his 'kids', I always thought that maybe some Mist that was stagnating in the more enclosed areas of the world could have been used to make more Vivi's. I not 100% sure, but I don't think Black Mages were made with the ability to sexually reproduce. I think they had to be made with one of the Mist machines, which I think Vivi used to make his mini-mes. But this part is only my take on it.