Whether or not we have a vector space depends on how you interpret "quadratic function whose graph passes through the origin."
In order for us to have a vector space, we will need, for example, to think of $3x$ as a quadratic function. For certainly $x^2+x$ is a quadratic function that passes through the origin, as is $-x^2+2x$. So if we are to have closure under addition, the function $(x^2+x)+(-x^2+2x)$, that is, $3x$, will have to be in the collection.
Indeed the identically $0$ function has to be in our collection, for two reasons.
If we do not allow it, then closure under addition can fail, since $(x^2-x)+(-x^2+x)=0$. Closure under multiplication by the constant $0$ also fails.
It is not difficult, however, to show that the set of functions of the form $ax^2+bx$, where $a$ and $b$ range over all the reals, is a vector space over the reals. Quite a number of axioms need to be verified, but each verification is easy.
As to your question about the "passing through $0$" part, one can also show that the set of all polynomial functions of degree $\le 2$ also is a vector space over the reals. It just is a different vector space than the one under consideration.
Remark: $1.$ In answering the homework question, it may be useful to be cautious. I would suggest doing the following: (i) Observe that if we interpret "quadratic" as meaning that the coefficient of $x^2$ is non-zero, then we do not have a vector space. (Of course one should explain why.) (ii) Show in detail that the collection of all functions of the shape $ax^2+bx$ is a vector space. Many of the steps can be possibly omitted. The key properties that have to be verified are closure under addition and under multiplication by scalars.
$2.$ You asked how to describe the vectors. Might as well use the standard polynomial notation, as in the answer above. The "pass through the origin" part just means the constant term is $0$.
The key word here is must. I.e., the statement claims that every system of linear equations with more equations than unknowns is inconsistent. That’s false. For example, the system $$\begin{align}
x &= 1 \\
2x &= 2
\end{align}$$ has two equations and one unknown, but is clearly consistent.
Best Answer
Planes are infinite in their extents. If we're given a vector $\mathbf{u}$, then the vector $k \mathbf{u}$ is obtained just by scaling the length of $\mathbf{u}$, but keeping the same direction. So, if the vector $\mathbf{u}$ lies in a plane $W$, then any vector $k \mathbf{u}$ will also lie in this same plane, no matter how large $k$ is.
The picture in your book is misleading -- it shows the plane $W$ as a bounded parallelogram, which is wrong. I guess it's a somewhat understandable mistake -- it's difficult to draw infinite things on the page of a book. But they could have drawn a region with a fuzzy border to indicate that it's infinite.