The noun "Noi" is not necessary. It is implicit by the conjugation of "scriem."
Thats true, however, this question repeats itself in "train", and for people like me who started learning just days ago, its better this way so it reminds which person exactly it is about :)
Why not putting -ul at the end of the word book because it is the book ?
Because "cartul" does not make sence in romanian. "Cartea" makes more sence.
I hope this helps :)