For approaches of (2) i.e. resolution of systems with coefficients in a division ring (a skew field), you can consult
[1] A. Heyting, Die Theorie der linearen Gleichungen in einer Zahlen- spezies mit nichtkommutativer Multiplikation, Math. Ann., 98, 465- 490 (1927).
[2] A.R. Richardson Simultaneous linear equations over a division ring. Proc. Lond. Math. Soc., 28, 395-420 (1928).
Now, you have also the non-commutative determinants of Gelfand and Retakh
arXiv:math/0208146
you will find in there a nice historical introduction on the subjecct of noncommutative determinants as well as a construction of the free division ring.
and in their book. Applied later to the theory of noncommutative symmetric functions started by Gelfand, Krob, Lascoux, Leclerc, Retakh, Thibon (up to my knowledge continued through seven papers).
A warning about the rank Let $\Gamma=F(a,b)$ be the free group on two letters and order it with a total (group) ordering as it can be done this though series, see e.g.
G. Duchamp, J.-Y. Thibon, Simple orderings for free partially commutative groups , International Journal of Algebra and Computation 2 No.3 (1992).
Then consider the skew field $\mathbb{Q}((a,b))$ (Malcev Neumann series, for example as in arXiv:math/0405133) which is the set of functions $\Gamma \rightarrow \mathbb{Q}$ with well-ordered supports (and usual operations). Then, the matrix $$ \begin{pmatrix} ba & a\\ b^2 & b \end{pmatrix} $$ has its columns left proportional but not right proportional. So, the vector space generated by the columns on the left has dimension one and on the right has dimension 2.