Is the fixation rate always equal to the mutation rate for neutral alleles?
The answer to your headline question is that no, fixation rate is not always equal to the mutation rate for neutral alleles. For instance:

Fixation rates for neutral alleles are affected by changes in population size, given a constant mutation rate. In general, fixation rates are lower in growing populations (Waxman 2012). 

This makes instinctive sense if you consider the single-generation fixation probability of a neutral allele which only exists at all but one copy in the population, i.e., the common allele (A) exists at 2N(t)-1 copies, whereas the less common allele (a) exists at 1 copy. To simplify, let's assume complete random mating. Under negative population growth (say N(t+1) = 0.9N(t)), there will be 0.9*(2N) places that the rare allele could possibly exist at t+1, and it has a 1/(2*N(t)) chance of filling each of them. Under positive population growth (say, N(t+1) = 1.1*N(t)), the rare allele could potentially occupy 1.1*(2N) places at t+1, but still has 1/(2*N(t)) chance of filling each of them. It is clear in this case that A has a reduced generation-wise likelihood of fixation under positive population growth than under negative population growth, simply because the rare allele has an increased chance of doggedly persisting. The finding applies to any allele near fixation.

If I have understood Kim and Stephan (2000) correctly, then overall fixation rates are also reduced by background selection. The main mechanism for this is that background selection purges linked neutral alleles, decreasing overall heterozygosity and thereby reducing the chance that a new, rare allele will become established. The same surely applies to selective sweeps, given linked neutral alleles.