Are there any archaeological evidences suggesting the existence of the Book of Mormon? The Old Testament has many archaeological findings which confirm its existence many years ago. A good example would be the discovery of Dead Sea Scrolls with some of the scrolls estimated to be created around 300 years before Christ.

The New Testament also has many archaeological proof of existence. One good example would be Codex Sinaiticus estimated to be created about 350 years after Christ.

I have no idea about the archaeological evidence for the existence of Book of Mormon at least some 1000 years ago. Is there any discovery till date which suggests the existence of the Book of Mormon before Joseph Smith? Did any ancient scholar mention it? Was it ever quoted by ancient writers? Such an important book about Jesus Christ should leave some traces of its existence in the past.

(Please note that this question is not asking for the evidence of events in the Book of Mormon)
To answer your question, I would challenge you to learn as much about the American continent as you can for the date that the Book of Mormon was compiled (~400AD). You will quickly discover that there are no other surviving records from that time period at all.1 In fact, the oldest manuscript written in the Americas known by historians–the Dresden Codex–was written half a millennia after Mormon compiled the Book of Mormon, and is one of only 15 ancient Mesoamerican records left in existence. 

The Spanish conquest of Mesoamerica erased most of the history there. Temples were torn down and Cathedrals were built upon their foundations. The vast majority of what we know about those cultures was written by their conquerers. 

If there were any codecies that supported the existence of the Book of Mormon, they would have likely been destroyed around the same time all of the Christians in Mormon's time were slaughtered, or during one of Bishop Diego de Landa's purges, and had the Gold Plates or any other plates made of gold been discovered, they would have been melted down into ingots, just like Atahualpa's ransom, and sent back to Spain. 

So no, there are no evidences comparable to the Dead Sea Scrolls, or Codex Sinaiticus that serve as evidence of the Book of Mormon's existence. It is unknown exactly how the scriptures were even shared with the people other than by the kings and prophets preaching to them. Mormon states in his book–The Words of Mormon–that the plate records he used to compile the Book of Mormon, which are briefly explained here, were handed down by the kings until they were delivered to him, who then spent a large portion of his life compiling them into the book which he delivered to his son Moroni, who finished the book, buried it, and 1400 years later revealed to Joseph Smith in resurrected form. 



1 List of Existing Codecies of Mesoamerican Literature.