Edited: to reflect the correct definitions.
Question 1: Why are the guts well-defined?
Answer 1: By the JSJ theory there is a unique collection of $I$-bundles (and Seifert fibered spaces) that contain (up to isotopy) all essential product disks and product annuli (and all essential tori). We throw away the $I$-bundles. Note that any Seifert fibered pieces remaining are pared solid tori. (There are no interesting Seifert fibered pieces because the original manifold $M$ has "non-degenerate" Thurston norm. So all essential tori in $M$ are parallel to boundary components.)
Question 2: Why is the pared guts well defined?
Answer 2: For exactly the same reason.
Question 3: What is the relation between pared guts and the sutured guts?
Answer 3: Pared guts are directed at understanding geometry. Sutured guts have an additional homological condition (the assumption of tautness).
Question 4: Why do we need to consider product disks in the sutured case, but not in the pared case?
Answer 4: We don't need them in either case. Consider the frontier of a regular neighbourhood of the union of a product disk and the sutures it meets.