For question 2, the answer is no. There are plenty of distinct knots (and hence having distinct fundamental group) with the same volume. For example, the Conway and Kinoshita-Terasaka knots differ by a mutation, and hence their complements have the same volume. You can get arbitrarily many knots with the same volume by doing mutations on knots that contain many Conway spheres.

For question 2, the answer is no. There are plenty of distinct hyperbolic knots (and hence having distinct fundamental group) with the same volume. For example, the Conway and Kinoshita-Terasaka knots differ by a mutation, and hence their complements have the same volume. You can get arbitrarily many knots with the same volume by doing mutations on knots that contain many Conway spheres.