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).