tl;dr> This clip from the end Raiders of the Lost Ark probably captures the essence of what happened rather well - it's not that it was "lost" so much as ignored and forgotten.
King Josiah ruled Judah over a period in which the re-discovery of YHWH overall was in vogue. For the past 77 years, two very wicked kings had ruled - Manesseh and his son Amon.
Of Mannesseh, it is said:
and he did what was evil in the sight of the Lord, yaccording to the despicable practices of the nations whom the Lord drove out before the people of Israel. For he rebuilt the high places zthat Hezekiah his father had destroyed, and he erected altars for Baal and made aan Asherah, bas Ahab king of Israel had done, cand worshiped all the host of heaven and served them. And he built altars in the house of the Lord, ... And he burned his son as an offering* and used fortune-telling and omens and dealt with mediums and with necromancers. He did much evil in the sight of the Lord, provoking him to anger.
The wickedest king of Israel presided over a huge decline in the worship of God. In such a regime, especially if new altars were being built in the Temple, it would not at all be surprising if the Book of the Law was "filed away" and forgotten.
And, unlike today, when book ownership is prevelant, mis-filing one copy really could prevent anyone from knowing about it. Books and Scrolls were, in antiquity, extremely expensive things. Even for an epistle as short as Romans, Raymond Brown has estimate that the parchment and ink alone would have cost $3000 - a fortune in those times.
Even today, "discoveries" are being made, not out in the field, but in museum vaults! Mere possession of an object, and knowledge that one has it, are two completely different things. Just look inside the drawers of your bathroom sink (if you live in the West) if you don't believe me.
So, to answer your questions directly:
Was the Book of the Law lost for a long time that King Josiah never saw it before?
It had been ignored for up to 77 years - after the days of Hezekiah, but during the 55 year reign of Manessah and 22 years of Amon's reign.
Does this discovery of the Book of the Law mean that the location of the book was not known before?
Whether or not it was "known" is debatable. It was in the Temple and always had been. But imagine trying to find a single artifact in a large building that served as a storehouse. (Remember, people didn't enter the Temple on a regular basis - only the priests did. It wasn't a church!)
Which copy of the Book of the Law was it? Was it the copy for the King or from the Levitical priests? (Deuteronomy 17:18)
There is nothing to suggest that it was anything other than the copy in the Temple. The copy for the King would definitely not have been kept, as Josiah's two predecessors did not belong to the cult of YHWH.
Did their ignorance of the presence of Book of the Law mean that nobody at that time cared to go inside the Temple to read from the Book of the Law?
Yup. (At least from the priest's point of view.) Remember, ordinary people were not allowed inside the Temple. Indeed, women weren't even allowed in the inner courts. Only the Priests had access to the interior storehouses, and only the Chief Preist had access to the Holy of Holies.
How may we refute people who say that someone forged the book and put it there?
Scholars can't and don't even try. Indeed, the "JEDP" hypothesis suggests that while the text itself formed orally over a period of 400 years (from roughly 1000 BC until 600BC), claims such as "Josiah being the greatest king that ever ruled Israel" suggest that the form we have today was last edited during his reign. That said, ultimately, even a biblical perfectionist such as myself understands that the Bible we have today is the one that God intended for us to have. This is the doctrine known as 'preservation.' Whether or not it has been edited is of little consequence - the church has done quite well with the Bible it has.