Quadratic forms and orthogonal basis

Question

I have the quadratic form $q(x,y,z) = x^2 - 2y^2 +xz +yz$

1. Give the polar form and the matrix of q in canonic basis. I think this question is ok : $$ A = \begin{pmatrix} 1 & 0 & 1/2 \\ 0 & -2 & 1/2 \\ 1/2 & 1/2 & 0 \end{pmatrix} $$

$$ f(u,u') = x x'-2 y y' + \frac{1}{2}(xz' + x'z) + \frac{1}{2}(yz' + y'z)$$

2. Find an orthogonal basis for q. I am not 100% sure of my answer : If first used Gauss algorithm and found $$ q(x,y,z) = \left(x+\frac{1}{2} z \right)^2 - 2\left(y-\frac{1}{4}z \right)^2 - \frac{1}{8}z^2 $$

Then I deduced the matrix $$ B= \left( \begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix}, \begin{pmatrix} 0\\ 1\\ 0 \end{pmatrix}, \begin{pmatrix} -\frac{1}{2}\\ \frac{1}{4}\\ 1 \end{pmatrix} \right) $$

3. Check if B is orthogonal finding the q matrix in the basis B from A: do I have to do a diagonalization?

Could q matrix in basis B be $\begin{pmatrix} 1 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & -1/8 \end{pmatrix}$ ?

4. Give a basis for q orthogonal in the plane generated by the two first vectors of canonic basis. I am still looking how to start this question.

Thanks for your help.