Yes, they are using Future Continuous because that's a direct equivalent of the Polish construction - exactly because that's a 'niedokonany' verb.
I used to think that dokonany/niedokonany translates quite directly into English, but it's not that easy. It's a personal nightmare for me ;) Often Future Simple seems to be a correct translation of niedokonany as well. "We will write a letter" sounds to my Polish ear as if we were talking about finishing that letter, but perhaps it can work just for the action as well..
Well, technically the word "dzieci" (children) is grammatically 'not masculine-personal plural', so "The children will be writing a letter" would be "Dzieci będą pisały list", true.
However, for this specific sentence, imagine that a child is saying that. The child knows its (his/her) own gender, as well as the genders of the other children that are supposed to work on this letter. Therefore the word 'child' is out of the picture, because every single child is considered either a boy or a girl, so they will know whether to use "pisali" (at least one boy) or "pisały" (no boys).
So while in theory you're right, in fact it has to be women/girls.
Although will is in common use in all forms shall is properly used in the first person singular unless the intent is to communicate a sense of determination or obligation then will is used in the first person and shall in the second and third persons. For English learners will is always acceptable but perhaps it is worth knowing about shall. 'Shall we dance?'