Those are too many questions for one post.
1- The field of $5$ elements can't be a vector space over the field of $3$ elements with its usual addition. That's because in any vector space $E$ over $GF(3)$, for every $e\in E$, it must be $e+e+e=0$ (because it's the same as $3e$, and $3=0$ in that field). That doesn't happen in $GF(5)$.
2- The argument I used before shows that it is necessary that $GF(p) $ and $GF(q) $ have the same characteristic. The characteristic of a field is the smallest integer $n$ such that the unity summed with itself $n$ times is zero: $1+1+\dots1=0$, in that field.
The characteristic of a field with $p^n$ elements, with $p$ prime, is equal to $p$. So, it's necessary that $p$ and $q$ are powers of the same prime number: the characteristic of both fields.
There is another necessary condition: take $q= t^k$, $p=t^l$. If $GF(p)$ was a vector space over $GF(q)$ say, of finite dimension $n$, then $p= q^n$, and that can only happen when $t^{kn}= t^l$, that is, when $l$ is a multiple of $k$.
3- This question only makes sense when the three fields are vector spaces over the same field $GF(r)$. And it's not clear what do you mean by "some field is a subspace of another field". Are you assuming that these subfields are included one into the other? In what way? Perhaps you want to know when one is a field extension of the other?
