Some techniques that were educational or helpful:

4.3.1:
The hint doesn't seem to help much here.

4.3.5:
The two projections sum to I, therefore you only need to compute one probabily, using the fact that probabilities sum to 1.

4.4.1:
A quick proof arises from using the definition of positive semi-definiteness.  Identify the same vector multiplying \rho on left and right.

4.4.4:
Pauli matrices anticommute.

5.1.2:
Use Schmidt decomposition

5.2.6:
Notice that the channel to the environment is also an erasure channel.  The input qubit either goes to the channel output or to the environment.

5.2.12:
There are some mistakes in the problem statement.  The trace on the RHS should be over E, and a either a not-gate is needed or a different isometric extension must be used than the one in Exercise 5.2.11.

5.4.1:
Use the square-root of \Lambda in the construction of the coherent measurement, and note that sqrt(\Lambda) >= \Lambda.