If $L(\Bbb{R}^n)$ is the space of $\Bbb{R}$-linear maps from $\Bbb{R}^n$ to $\Bbb{R}^n$, which of the following are true?

(3) is certainly true (hence (4) is false). For example, you can take a transformation $T$ whose matrix representation (in whatever basis you like) is $$ \begin{pmatrix} 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0 \end{pmatrix}. $$

It is correct that (2) is true (hence (1) is false). Indeed, (2) would be true if $\Bbb R^5$ were replaced by $\Bbb R^{2k+1}$ for any nonnegative integer $k$. Can you see why?