Ravenel in his *Complex Cobordism and Stable Homotopy Groups of Spheres* attributes this result to Stong, in *Determination of $H^*(BO(k,⋯,∞),Z_2)$ and $H^∗(BU(k,⋯,∞),Z_2)$*, but looking at that paper (which is concerned mainly with the "unstable" cohomology of the various constituent spaces of $ko$), he attributes this further to

*Adams, J. F.*, On Chern characters and the structure of the unitary group, Proc. Camb. Philos. Soc. 57, 189-199 (1961). ZBL0103.16001.

There you can find indeed the required result as Lemma 4.

Regarding your second question, the cohomology $H^*(KO;\mathbb{F}_2)$ is zero for chromatic reasons (the spectrum $KU\wedge H\mathbb{F}_2$ carries an isomorphism of the additive and multiplicative formal group law in characteristic two, so it must be trivial, and then you can use the fact that $KU=KO/\eta$ to conclude, since $\eta$ is nilpotent in $\pi_*KO$), hence the map $H^*(KO;\mathbb{F}_2)→H^*(ko;\mathbb{F}_2)$ is trivial.