Unless I'm mistaken, all statements are wrong. The first one is wrong because it implies $\left|\lambda\right|=1$ if $T$ is nonzero, which is not true because you can let $T=2\mathrm{Id}$. (Perhaps you formulated this statment wrongly? Maybe it should be $\left\Vert T(x)\right\Vert =\left|\lambda\right|\left\Vert x\right\Vert$ ?).
The second statement is also wrong: definte $T$ to be $(x,y)\mapsto(\sqrt{2}x,0)$; then $\left\Vert (1,1)\right\Vert =\sqrt{2}=\left\Vert (\sqrt{2},0)\right\Vert =\left\Vert T((1,1))\right\Vert$ but in this case $\lambda=\sqrt{2}$ (because $T^{2}=\sqrt{2}T$).
The third one is not correct because you can take $T$ to be $(x,y)\mapsto x$.
The fourth statement is wrong because you can let $T=2\mathrm{Id}$, which is not singular. This is essentially the only possibility, though: if we assume $T$ is invertible, then clearly $\lambda\neq0$, so we can set $S=\frac{1}{\lambda}T$, which is also invertible (with inverse $\lambda T^{-1}$). We have $S^{2}=\frac{1}{\lambda^{2}}T^{2}=\frac{1}{\lambda^{2}}\lambda T=\frac{1}{\lambda}T=S$, so $S=\mathrm{Id}$, i.e. $T=\lambda\mathrm{Id}$.