While पुस्तक and किताब are synonyms, पुस्तक is a more 'formal' word and is not used as much in day-to-day conversations. But it should definitely be an accepted response to this question.

well that word is of "pure Hindi" spoken in king times. Pure hindi was a direct translation from Sanskrit. However modern hindi was created for changing circumstances. That has been replaced with modern hindi which doesn't use that word and additionally combines English into modern hindi. Make sense.

किताब is a feminine noun. मेरा/मेरी takes the gender of its object not the person who is speaking.

There is no foolproof way to know the gender of a noun by looking at it. You can sometimes guess the gender based on things like word ending (words ending with ा are more likely to be male, those ending in ी are more likely to be female etc) or the category of the word (country names are usually male, rivers are usually female etc) but there are a lot of exceptions to each of these rules. The best way is that whenever you encounter a new Hindi word, you learn it with its gender. That is, instead of learning कुत्ता='dog', बिल्ली='cat', you learn मेरा कुत्ता='my dog', मेरी बिल्ली='my cat'.