The story is told of a man named Bishr who was coming home drunk one rainy night and who saw apiece of paper with some writing on it in the mud. Upon closer examination he saw that the writing was the name of God in Arabic, ALLAH. He picked up the paper, cleaned it, and placed it in a high place in his house. Later on he had a dream in which he heard Allah tell him: Because you raised our Name high, We shall raise up your name in this world and the next. Bishr became a great saint who is known even to many non-Muslims. You can read his story in Qushayri's "Sufi Principles" available in English, and Attar's "Memorial of the Saints" published by Penguin books.
Regarding your ownership of the Qur'an, it is not an accident but an invitation by the Creator to accept Him into your heart as you have accepted His book into your home. That you are invited is confirmed by your question, since another person would not care, but you care. Therefore I confirm the extended hand of my God and your God to you by inviting you to believe in Him, and in His Prophet Muhammad as well as in all His Prophets. If you do, then that Holy Book and what it contains will change you for the best and increase your sense of everything sacred and holy. So, in answer to your question about handling, the first recommended act is: to believe. The second is: to keep the book in a clean elevated spot, not to place anything on top of it, to hold the book with respect, not to fling it or let it drop, and to love it and hold it dear.
These are particular verses or sets of verses:
The first chapter [the Opening] The first verses of every single chapter The verse of the Throne 2:255 The last two verses of the second Sura 2:285-286 Surat al-Taghabun (#64) Ikhlas (#112)
I would like to recommend to you the following suras:
Al `Imran (#3) Maryam (#19)
And I wish you guidance to what is best with sincere regards,