Find a basis for orthogonal complement in R⁴ 
How do I approach part 2? I found the projection of 1. to be (6,-2,2,-2) but what do I do now?

Alternatively, one could solve the linear system $Ax=0,$ where
$$A=\begin{bmatrix}1&2&3&4\\2&5&0&1\end{bmatrix}.$$
$A$ us equivalent to the matrix
$$A=\begin{bmatrix}1&0&15&18\\0&1&-6&-7\end{bmatrix}$$
which has the solution set $\operatorname{span}\{(-15,6,1,0),~(-18,7,0,1)\}.$
Note that since $\mathbb{R}^{4}=W\oplus W^{\perp}$ and $\operatorname{dim}W=2,$ you get that $\operatorname{dim}W^{\perp}=2,$ as well.
