The short answer to your question about Specht modules for other types than the symmetric group is "yes". The long answer is that you have to dig into the extensive literature built up around cyclotomic Hecke algebras and the like. One place to look is the arXiv, where the papers of Andrew Mathas (Sydney) and his collaborators from recent years can be found. A recent example is here, but there are many related papers by other people including Meinolf Geck and his collaborators.

Though I haven't followed all of these developments closely, my perception is that the emphasis tends to shift from the finite Coxeter groups such as $S_n$ to their Iwahori-Hecke algebras and related module categories. Combinatorics is a major subtheme throughout, as well as categorification.