To correctly match LIST-I with LIST-II, we need to understand the specific scientific concepts named in LIST-II and identify which phenomenon or rule from LIST-I they describe.

1. A. Shorter appendages in mammals living in cold climates than those living in warm climates: This is described by Allen's Rule which states that endotherms from colder climates usually have shorter limbs or appendages compared to those from warmer climates. This helps reduce heat loss. Therefore, A matches with III.
2. B. Shorter and rounder bodies in mammals living in cold climates than those living in warm climates: This fits Bergmann's Rule, which postulates that individuals of larger size are found in colder environments, and smaller sizes in warmer regions. Larger bodies have a smaller surface area relative to volume and are better at conserving heat. Thus, B matches with IV.
3. C. A sustained directional shift in a population's average characteristics: This describes Anagenesis, which involves evolutionary change within a lineage. This leads to a shift in average characteristics over time. Hence, C matches with I.
4. D. Speciation through a branching mechanism: This is associated with Cladogenesis, which is the process of species formation through divergence from a common ancestor, resulting in a branching evolutionary tree. Thus, D matches with II.

Therefore, the correct answer is: A-III, B-IV, C-I, D-II.