Edited: to reflect the correction from the comments.

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. We further decompose the Seifert fibered pieces to get pared solid tori  (which themselves may be thrown away if they are products).

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.

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.

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). This is exactly what we are going to cut out and throw away.

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.

Edited: to reflect the correction from the comments.

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. We further decompose the Seifert fibered pieces to get pared solid tori (which themselves may be thrown away if they are products).

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.