Quantum PAC learning
I'll take a shot at your first question:

Are there examples of natural function families that quantum computers can learn faster than classical computers given cryptographic assumptions?

Well, it depends on the exact model and the resource being minimized. One option is to compare the sample complexity (for distribution-independent PAC learning) of the standard classical model with a quantum model that is given quantum samples (i.e., instead of being given a random input and the corresponding function value, the algorithm is provided with a quantum superposition over inputs and their functions values). In this setting, quantum PAC learning and classical PAC learning are basically equivalent. The classical upper bound on sample complexity and the quantum lower bound on sample complexity are almost the same, as shown by the following sequence of papers:

[1] R. Servedio and S. Gortler, “Equivalences and separations between quantum and classical learnability,” SIAM Journal on Computing, vol. 02138, pp. 1–26, 2004.
[2] A. Atici and R. Servedio, “Improved bounds on quantum learning algorithms,” Quantum Information Processing, pp. 1–18, 2005.
[3] C. Zhang, “An improved lower bound on query complexity for quantum PAC learning,” Information Processing Letters, vol. 111, no. 1, pp. 40–45, Dec. 2010.

Moving on to time complexity, using the same quantum PAC model, Bshouty and Jackson showed that DNFs can be quantum PAC learnt in polynomial time over the uniform distribution [4], further improved in [5]. The best known classical algorithm for this runs in $O(n^{\log n})$ time. Atici and Servedio [6] also show improved results for learning and testing juntas.

[4] N. Bshouty and J. Jackson, “Learning DNF over the uniform distribution using a quantum example oracle,” SIAM Journal on Computing, vol. 28, no. 3, pp. 1136–1153, 1998.
[5] J. Jackson, C. Tamon, and T. Yamakami, “Quantum DNF learnability revisited,” Computing and Combinatorics, pp. 595–604, 2002.
[6] A. Atıcı and R. Servedio, “Quantum Algorithms for Learning and Testing Juntas,” Quantum Information Processing, vol. 6, no. 5, pp. 323–348, Sep. 2007.

On the other hand, if you're interested in the standard classical PAC model only, using quantum computing as a post-processing tool (i.e., no quantum samples), then Servedio and Gortler [1] observed that there is a concept class due to Kearns and Valiant that cannot be classically PAC learnt assuming the hardness of factoring Blum integers, but can be quantumly PAC learnt using Shor's algorithm.
The situation for Angluin's model of exact learning through membership queries is somewhat similar. Quantum queries can only give a polynomial speedup in terms of query complexity. However, there is an exponential speedup in time complexity assuming the existence of one-way functions [1].
I have no idea about the second question. I'd be happy to hear more about that too.