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${}^\dagger$Added 9/2/16: I realize now that what I wrote below doesn't answer Question (1). It supplies an argument for Question (2).

Added 9/7/16: I just got access to the paper:

James, I. M. On the iterated suspension. Quart. J. Math., Oxford Ser. (2) 5, (1954). 1–10

which is an explicit reference to Greg's questions on the level of homotopy groups. See section 2.

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${}^\dagger$Added 9/2/16: I realize now that what I wrote below doesn't answer Question (1). It supplies an argument for Question (2).

The answer is yes.

${}^\dagger$Added 9/2/16: I realize now that what I wrote below doesn't answer Question (1). It supplies an argument for Question (2).

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The answer is yes.$^\dagger$

Consider the commutative diagram $$ \begin{CD} ``?" @>>> F_\ast(S^{n}_+,S^{n}) @>E^\sharp>> \Omega^{n+1}S^{n+1} \\ @VVV @VV E V @| \\ \Omega \Sigma S^{n} @> i>> F_\ast(S^{n}_+,\Omega \Sigma S^{n}) @>R>> \Omega^{n+1}S^{n+1} \\ @V (a) VV @VV HV @VVV \\ F_\ast(S^{n}_+,\Omega \Sigma S^{2n}) @= F_\ast(S^{n}_+,\Omega \Sigma S^{2n}) @>>> \ast \end{CD} $$ where the horizontal maps are fiber sequences, with the homotopy fibers of the first two rows taken at $1$. We need to identify ``?'' in the $2n$ range (more precisely, we need to show that it is about $(2n)$-connected.

Consider the commutative diagram $$ \begin{CD} \text{"?"} @>>> F_\ast(S^{n}_+,S^{n}) @>E^\sharp>> \Omega^{n+1}S^{n+1} \\ @VVV @VV E V @| \\ \Omega \Sigma S^{n} @> i>> F_\ast(S^{n}_+,\Omega \Sigma S^{n}) @>R>> \Omega^{n+1}S^{n+1} \\ @V (a) VV @VV HV @VVV \\ F_\ast(S^{n}_+,\Omega \Sigma S^{2n}) @= F_\ast(S^{n}_+,\Omega \Sigma S^{2n}) @>>> \ast \end{CD} $$ where the horizontal maps are fiber sequences, with the homotopy fibers of the first two rows taken at $1$. We need to identify "?" in the $2n$ range (more precisely, we need to show that it is about $(2n)$-connected.