Regarding question 11):
a) In addition to the potentials for which all bound orbits are closed there are further central force potentials with analytic solutions, please cf.
http://en.wikipedia.org/wiki/Classical_central-force_problem#Central_forces_with_exact_solutions (Sorry, not quite what you were asking for [I read too fast], but perhaps interesting nevertheless.)
b) Perhaps this paper on "Multiparticle systems with a particle-interaction potential homogeneous of degree α = −2" by Borisov et al. is interesting for you: http://ics.org.ru/doc?pdf=1359&dir=e. ("Some new integrable and superintegrable systems generalizing the classical ones are also described.").