I don't think any of the statements in the problem hold true, since a counterexample can be found for each one. For (1), let
$T = \left( \begin{matrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{matrix} \right)$
Then $T^2 = 6T$ and $T:\mathbb{R}^n \rightarrow \mathbb{R}$. However,
$e_1 = \left( \begin{matrix} 1 \\ 0 \\ 0 \\ \end{matrix} \right), e_2 = \left( \begin{matrix} 0 \\ 1 \\ 0 \\ \end{matrix} \right) $
produces $T(e_1) = e_1, T(e_2) = 2e_2$. So (1) is not true. Similarly, (3) is not true since $T^2 \ne \lambda I$ for any $\lambda\in\mathbb{R}$. For statement (2), let
$ T = \left( \begin{matrix} 2 & 1 \\ 0 & 0 \\ \end{matrix} \right), e_2 = \left( \begin{matrix} 0 \\ 1 \\ \end{matrix} \right) . $
Then $T^2 = 2T$, but $||T(e_2)|| = ||e_2||$, disproving (2). Finally, you can disprove (4) by considering the matrix $T = \left(2\right)$, which is nonsingular. Hence, all statements are false.