for example, if we are given 2-form $\omega=2xdx\wedge dy+2ydy\wedge dz$, what are the points where the form vanishes? I can only think of points $(0,0,z)$, is it all？
Additionally, if we have a form a on manifold where it does not vanish, when we pull back the form by $\alpha^*$ which is corresponding to the coordinate patch at the point, what can we say about the k-form that we pulled back? Is it still nonvanishing?
It really depends on what you mean by the vanishing of $\omega$ at a point $p$. Do you mean that $\omega(p)(v,w) = 0$ for all $v,w\in\Bbb R^3$, or do you wish to find a submanifold $\alpha\colon M\hookrightarrow \Bbb R^3$ with the property that $\alpha^*\omega = 0$ (i.e., $\omega(p)(v,w)=0$ for all $p\in M$ and $v,w\in T_pM$)? For example, consider the surface $M=\{x^2-2yz=\text{constant}\}$. Then $\alpha^*\omega = 0$ on $M$.