Some techniques that were educational or helpful:

10.8.1:
Decompose the measurements (spectrally) into rank one measurements, indexed by an additional variable.  The original measurement is the one obtained by discarding the new measurement outcome.  The data processing inequality completes the proof.

11.6.3:
Use Theorem 11.5.2.

11.7.2:
Use (11.58), which uses Theorem 11.2.2.

11.7.4:
Construct a classical-quantum state.

11.8.2:
Use Exercise 11.8.1 (better yet, show it yourself).

11.9.1:
It's best to express the measurement (accessible information) as the quantum mutual information of a coherent measurement.

12.4.2:
Optimized by maximally mixed state.  Use concavity to claim this.

12.4.3:
Again optimized by maximally mixed state.  This can again be shown using concavity, or mutual information can be reduced for this channel to a constant times the input entropy.

12.5.5:
Same steps as proof of additivity of mutual information (Theorem 12.4.1) except one inequality is not needed.