Concerning your first question, I think a good place to learn about it is Markus Banagl's book "Topological invariants of stratified spaces" (chapter 6) or have a look at Paul Siegel's paper "$KO$-homology at odd primes" Amer. J. Math. 105 (1983), 1067-1105 (in particular the appendix where Sullivan's approach is explained).
In Siegel's paper you have a bordism theory of some singular spaces called Witt spaces. Such spaces carry rational L-classes and this bordism theory when tensored by $\mathbb{Z}[1/2]$ is naturally isomorphic to $ko_*(-)\otimes \mathbb{Z}[1/2]$.
Thus for any Witt space in particular for smooth manifolds you have a fundamental class in Witt bordism, when tensored over the rational you get Goresky-McPherson L-classes (obtained thanks to intersection cohomology) and thus in the smooth case "ordinary" L-classes. Then use the natural isomorphism between this bordism theory and $ko$ at odd-primes to get your formula.
Concerning your second question you can have a look at theorem 11.1 "Multiplicativity of the L-theory fundamental class" of this paper: "The L-homology fundamental class for IP-spaces and the stratified Novikov conjecture" by Markus Banagl, Gerd Laures and Jim McClure (available on Banagl's homepage).