I'm trying to understand the main idea of Atiyah's proof (https://arxiv.org/abs/1610.09366). Although there were discussions on MO year ago I couldn't find answers to my questions.

Consider the isomorphism on the 4th page: "$KSp(\mathbb{R}^6) \rightarrow KR^{(7,1)}(pt) = \mathbb{Z}_2$"

My questions are as follows:

1. Atiyah says that the $KSp(\mathbb{R}^6)$ is "the stable homotopy group determined by Bott". But as far as I understand $KSp(\mathbb{R}^6)$ is the K-theory of vector bundles over base space $B$, s.t. the isomorphisms from definition of vector bundle $\pi^{-1}(b) \mapsto \{b\} \times \mathbb{R}^n$ lie in sympletic group $Sp(n)$. How does it connected with each other? Or how should I think about $KSp(\mathbb{R}^6)$?

2)Moreover, after that Atiyah claims that $KSp(\mathbb{R}^6)$ is "just the reduced $KSp$ of $\mathbb{S}^6$". How could one see it?

3)Next, Atiyah says that "There are natural forgetful maps from complex K-theory to KR theory and in dimension 6 the integers go to 0, so a hypothetical complex structure on $\mathbb{S}^6$" would give an even element. But we know one almost complex structure J(0) which is odd..." Here I can't understand the meaning of "even and odd". In what terms should I talk about it?

I will be grateful if someone makes these things more clear to me. Anyway, is this proof accepted as true in math society?