1st Question: It will always be open after that quest. But, if you didn’t purchase it, you won’t see options to decorate it or clean up the mess. The dust is still part of the ambient background and most of the blood and old items are still present. You’ll have to do more quests around Windhelm for the Jarl to give you permission to get a house in the city.
(Also note: if you did the Dark Brotherhood quest at the beginning with the woman who wants you to kill her ex-fiance, it is impossible to get the house. It’s a stupid bug that’s never been fixed)
2nd Question: It’s a book in your inventory.