I think the answer to the last question is no. As far as I understand, the "general linear groups over the field with one element" are supposed to be the symmetric groups. Therefore the statement that the algebraic K-theory of the field with one element can be identified with the stable homotopy groups spheres is essentially the Barratt-Priddy-Quillen theorem, which says that the group completion of $\coprod_n B\Sigma_n$ is the sphere spectrum.
As far as I can tell, the Barratt-Priddy-Quillen theorem does not yield that much explicit information stable homotopy groups of spheres. On the positive side, I would like to mention that Smirnov made an attempt to give an explicit presentation of the $E_\infty$ page of the Adams spectral sequence as an $A_\infty$ algebra: "Description of stable homotopy groups of spheres in the language of A∞-algebras." (Russian) Uspekhi Mat. Nauk 51 (1996), no. 1(307), 171--172; translation in Russian Math. Surveys 51 (1996), no. 1, 171–172. But I don't know the details of this work.