There are Gröbner basis algorithms that work over $\mathbb{Z}$, e.g., can decide is some polynomials generate the unit ideal over $\mathbb{Z}$.

I believe the question of whether indecomposable matroids are minimal is open. Most the experts I've talked to believe that they need not be, but an example will be hard to find.

I'm not entirely sure what you mean by "corank subdivision," but in general the regular subdivision induced by lifting to height equal to the corank of another matroid does not in general give a subdivision into matroid polytopes (although this is true for the uniform matroid). See Theorem 5.2 in arxiv.org/pdf/1902.05592.pdf.

See the question mathoverflow.net/questions/452598/minimal-connected-matroids‌​, although it is a little different.