Questions about Definition of Vector Spaces
Question 1.
A general definition of a "product" is the following: suppose you have three vector spaces $X,Y,Z$ all over the same field. Then, a bilinear mapping $\beta:X\times Y\to Z$ is essentially what we mean by a "product". So you can certainly consider the notion of products/multiplication. But, what is important to note is that this is extra information that you have to provide; it is not part of the vector space axioms, hence there is no standard/canonical choice in general.
You certainly can look at vector spaces equipped with dot products (more commonly called inner products). More precisely, suppose $V$ is a vector space over $\Bbb{R}$. Then, an inner product on $V$ is a bilinear, symmetric positive-definite mapping $\langle\cdot,\cdot\rangle : V\times V \to \Bbb{R}$. By prescribing an inner product, we are in essence prescribing a geometry for the vector space $V$. If you want to generalize some more, we can weaken the conditions imposed. For instance, a bilinear, symmetric, non-degenerate mapping $g:V\times V\to \Bbb{R}$ is called a pseudo-inner product, and we refer to the pair $(V,g)$ as a pseudo-inner product space. (I intentionally left out the complex case since the definitions are slightly different). The notion of a cross product is more subtle so let me not go into the details; but really if you formulate an appropriate definition, you can define various types of "products".
Question 2.
If $\mathbb{F}$ is any subfield of $\Bbb{R}$, then we can consider $\Bbb{R}^n$ as a vector space over $\Bbb{F}$ (by restricting the "usual" operations to the field $\Bbb{F}$). The most obvious examples are $\Bbb{F}=\Bbb{R},\Bbb{Q}$.
Question 3.
Suppose $V,W$ are vector spaces over fields $F_1,F_2$ respectively. To define the notion of a linear map $T:V\to W$, we would like the following equation to hold for all $x,y\in V,\lambda\in F_1$: $T(\lambda x+y)=\lambda T(x)+T(y)$. Well, to talk about $\lambda T(x)$, we need the target space to be considered as a vector space over $F_1$, so we'd need $F_1\subset F_2$. If you want to talk about isomorphisms, then you would also need the inverse map to be linear, so we want $F_2\subset F_1$. Thus, we always consider them over the same field $F$.
Question 4.
If $V$ is a vector space over a field $F$, and $W\subset V$ is any non-empty subset, then of course, by definition, to say $W$ is a vector space over $F$ we have to verify a whole bunch of axioms. But, it's a nice fact that we actually don't have to do so much work: as long as we check $0\in W$ (or just that $W\neq \emptyset$), and that $W$ is closed under addition and scalar multiplication, then it follows that $W$ (equipped with the restricted addition and scalar multiplication as operations) is a vector space over $F$ in its own right (this is usually called the "subspace" criterion or something). The nice thing is all the axioms like associativity/commutativity or addition/ distributive laws etc all hold in $W$ because they hold in $V$ and since $W\subset V$.