The main results of Rado's Note on Independence Functions settle all three questions. The first few lines of Effective Versions of Two Theorems of Rado give a perfect recap of those results, so here they are verbatim:

Our starting point is given by the following two theorems of Rado [5].

 

Theorem 1 (Rado, 1957). Let $M$ be a matroid representable over a field $K$. Then $M$ is representable over a simple algebraic extension of the prime field of $K$.

 

Theorem 2 (Rado, 1957). Let $K$ be an extension field of $\mathbb{Q}$ of degree $N$ , and let $M$ be a matroid representable over $K$. Then there is a positive integer $c$ such that given any prime $p > c$ there is a positive integer $k = k(p) ≤ N$ such that $M$ is representable over $GF(p^k)$. For infinitely many $p$, $k(p) = 1$.

 

Together, these two theorems say that if a matroid is linearly representable, then it is representable over a finite field.

The main results of Rado's Note on Independence Functions settle all three questions. The first few lines of Effective Versions of Two Theorems of Rado give a perfect recap of those results, so here they are verbatim:

Our starting point is given by the following two theorems of Rado [5].

Theorem 1 (Rado, 1957). Let $M$ be a matroid representable over a field $K$. Then $M$ is representable over a simple algebraic extension of the prime field of $K$.

Theorem 2 (Rado, 1957). Let $K$ be an extension field of $\mathbb{Q}$ of degree $N$ , and let $M$ be a matroid representable over $K$. Then there is a positive integer $c$ such that given any prime $p > c$ there is a positive integer $k = k(p) ≤ N$ such that $M$ is representable over $GF(p^k)$. For infinitely many $p$, $k(p) = 1$.

Together, these two theorems say that if a matroid is linearly representable, then it is representable over a finite field.