Can we view a vector space as a field extention? We can view field extentions as vector spaces over a field (an idea that in my experience has not really been explained but I get it more or less).
But can any vector space over a field also be seen as an extention of that field? I think no, since vector spaces are not fields. And frankly I don't see how any $r$ dimension vector space $V$ over a field $K$ of the form $\{\alpha_1v_1+...+\alpha_rv_r\ :\alpha_i\in K, v_i\in B\}$ and $B$ is some $K$-basis of $V$ can be a field...

To your first question, definitely yes. Number fields are exactly this kind of object: if you adjoin $\alpha \in \mathbb{C}$ to $\mathbb{Q}$, written $\mathbb{Q}(\alpha)$, then $\mathbb{Q}(\alpha)$ is a vector space over $\mathbb{Q}$ with scalar multiplication defined as in $\mathbb{Q}(\alpha)$. The basis for some extensions is easy to find: if $d$ is a cube-free integer different from $0$ and $1$, then $\mathbb{Q}(\sqrt[3]{d})$ as a vector space over $\mathbb{Q}$ has basis $1, \sqrt[3]{d}, \sqrt[3]{d}^{2}$.
As for your second question, I'd say no for a different reason. Consider the vector space $\mathbb{R}^{n}$ over $\mathbb{R}$ for $n \geq 3$. Then $\mathbb{R}^{n} = \mathbb{R} \times \mathbb{R} \times \cdots \times \mathbb{R}$ is not a finite extension of $\mathbb{R}$, it's the Cartesian product of $n$ copies of $\mathbb{R}$.
