"I am putting your book in my bag."
Translation:Ik stop jouw boek in mijn tas.
I think you mean liggen. In Dutch, when you put an object inside of another object, in this case a book inside of a bag, we mostly use stoppen in. Liggen is used in different circumstances, as you can read in this Dutch grammar section: https://www.duolingo.com/comment/5785064
No. The Dutch present continuous is only used for ongoing actions and would sound very odd in this sentence.
What do you mean by "ongoing"? Though we usually say that sentence when we are abount to put a book in a bag, one can very well be saying that he is putting a book in a bag while actually putting a book in a bag.