Are all vectors straight lines? Is there a math field that deals with quadratic, cubic etc. vectors? Or a non-linear equivalent of a vector? If so, why are they so much less common than linear vectors?

What you are considering to be a vector can be abstracted as follows: Place the line segment so that its one end is on the origin and the other end is somewhere in the Cartesian plane. The coordinates of the other end can be identified with this straight line and so may be called a vector. So for example $(2,3)$ is a vector.
This can be generalized to $3$ dimensional space, and in fact to $n$-dimensional space with the notion of a straight line joining the origin to the point. (You can abstract it even further in the idea of an arbitrary vector space.)
Now your question, if I am understanding it correctly, is as to why we don't consider curves instead of straight lines to geometrically represent the vector. We could, but that would unnecessarily complicate things: The curve would have to be carefully chosen so that it is defining an invertible function on $\mathbb{R}$ with an appropriate parameter to ensure all points could be represented. Straight lines are pretty simple to handle and satisfy this criteria and hence are used.
