I would like to learn this topic of algebraic topology but I cannot find a relevant reference to answer my basic questions on the subject (for example, is there a Hurewicz theorem for regular homotopy groups? under which conditions can one homotop a non-zero class in a non-trivial higher homotopy group to an immersion?).
The Smale-Hirsch theorem says that the existence of an immersion in a given homotopy class is equivalent to the existence of certain bundle map, in turn equivalent to the existence of a section of a certain bundle. See Hirsch, Morris W. Immersions of manifolds; the Mathscinet review (MR0119214)by Kervaire is a good summary. Thus these questions are `reduced' to algebraic topology, in particular to obstruction theory. This is of course a great thing, but I doubt there's an answer to your question in any great generality, because obstructions are generally hard to compute.
The best hope for a general answer would be in the so-called stable range where the dimension of the source is less than 2/3 the dimension of the target; see Haefliger, André; Hirsch, Morris W. Immersions in the stable range. Ann. of Math. (2) 75 1962 231–241.