Why do we strap the tefillin around our arm 7 times? Also, what is the significance of the way we wrap the tefillin around our finger?
7 times specifically comes from the Arizal. In Kabbalah, there are seven Sefiros which correspond to emotional attributes. Since the Tefillin of the hand are about the subjugation of the heart (i.e. emotions) these attributes are specifically represented through the 7 circles. (This isn't the only custom, but that is the point of this one).
The Tefillin around the finger are part of the system of wrapping which spells Hashem's name of Shin Dalet and Yud. The finger forms the Yud. How the other letters are formed is a matter of variant customs.