An evident general construction is to take any multiplicative cohomology theory $E^*$ and a cohomology operation $\psi$ in that theory, say taking $E^q$ to $E^{q+n}$. For an $E$-oriented real $q$-bundle $\xi$ over $X$, $\theta^{-1}(\psi(\mu))$ gives the value of a characteristic class, where $\mu$ is the Thom class of $\xi$ and $\theta$ is the Thom isomorphism. For stable $\psi$, defined for all $q$, this will give a characteristic class on all $E^*$-oriented bundles of the sort wanted, modulo precision about condition (ii). Not all characteristic classes of the sort wanted arise this way. For example, as proven by Novikov, the rational Pontryagin classes are homotopy invariant (condition (i)), and I think they should satisfy (ii).

An evident general construction is to take any multiplicative cohomology theory $E^*$ and a cohomology operation $\psi$ in that theory, say taking $E^q$ to $E^{q+n}$. For an $E$-oriented real $q$-bundle $\xi$ over $X$, $\theta^{-1}(\psi(\mu))$ gives the value of a characteristic class, where $\mu$ is the Thom class of $\xi$ and $\theta$ is the Thom isomorphism. For stable $\psi$, defined for all $q$, this will give a characteristic class on all $E^*$-oriented bundles of the sort wanted, modulo precision about condition (ii). [Senior moment nonsense eliminated]. You are studying the J-map $BO\to BF$ (or $BU \to BSF$), where $BF$ classifies stable spherical fibrations (oriented for $BSF$). A lot more is known than Adams knew. In particular, he didn't yet have the Adams conjecture. Rationally, $BF$ is a point. At an odd prime $p$, $BF$ factors as $BJ\times BCokerJ$, and at $2$ there is a non-split fiber sequence $BCokerJ \to BSF \to BJ$. The $J$-map at $p$ is best thought of as a map $BSpin\to BSF$ ($BO\simeq BSpin$ at $p>2$). By the Atiyah-Bott-Shapiro orientation, the $J$-map $BSpin\to BSF$ factors through the classifying space $B(SF;kO)$ for $kO$-oriented spherical fibrations and, at any $p$, $B(SF;kO)$ splits as $BSpin\times BCokerJ$ (but BSpin is seen in two pieces, one carrying the Wu classes, the other the rest of the Adams splitting). The intuition is that $BCokerJ$ and thus the unknown parts of the stable homotopy groups of spheres can be ignored, leaving the focus on the quite computable composite $BSpin \to BSF \to BJ$. This is too fast, and details are in Chapter V of $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra''. In ordinary mod $p$ cohomology, calculations are thoroughly understood but don't shed light on your question. They are also understood for $K$-theory, by work of Hodgkin and Snaith, and here the intuition that $Coker J$ can be ignored is made precise by Hodgkin's result that $\tilde K(BCokerJ)=0$.