Security of RSA variants with $e=n-2$ and $e=n$
Rather obvious, but does answer one of the questions: both can be reduced to the security of RSA with $e' = a_n n (n-2)$ for any (reasonably sized) public integer $a$ which can be a function of $n$. For example, given $c \equiv m^n$ for a random unknown $m$, we solve $c^{n-2} \equiv m^{n(n-2)}$ for $m$. With probability at least $1/gcd(e,\lambda(n))$ the answer will be $m$.