Considering T means taking derivatives, it must be corrected as

T:\mathbb{R}_2[x]\rightarrow \mathbb{R}_0[x]. But then we don't have the linear transformation in the form T:V\rightarrow V and therefore eigen values/vectors are not defined.

Eigen value is 0 and eigen vectors are spanned by \{\begin{pmatrix} 1\\0\\0\end{pmatrix},\begin{pmatrix} 0\\1\\0\end{pmatrix}\} for A

 A=\begin{pmatrix} 0 &0&2\\0&0&0\\0&0&0\end{pmatrix}

Eigen value is 0 and eigen vectors are spanned by \{1,x\} for T