How are characteristic classes morphisms of infinite loop spaces? (if they are)

Yes it is true. You have correctly interpreted the intended meaning of the phrase ``fibration of infinite loop spaces''. One early reference is chapter I of $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra, available at http://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf

Your sequences are all arise in the following standard way. Suppose $x$ is an $(n-1)$--connected spectrum and let $X = \Omega^{\infty} x$. One always has a fibration sequence $$y \rightarrow x \rightarrow \Sigma^n H\pi_n(X)$$ and applying $\Omega^\infty$ to this yields a fibration sequence of spaces $$Y \rightarrow X \rightarrow K(\pi_n(X),n).$$

$Y$ is the $n$--connected cover of $X$.

(In your situation, one has successive covers $bspin \rightarrow bso \rightarrow bo$.)