so let's say I got a Mycosynth Golem and 9 other artifacts. Does Reaper King effectively cost me zero mana since he has the alternative cost of generic mana? And in theory could I cast the King with half colored mana at my choice and half zero cost (due to affinity reduction) as well?
In general, for purposes of cost reductions such as affinity (C.R. 702.41a), while a player is casting a spell with hybrid mana symbols in its mana cost, that player "announces the nonhybrid equivalent cost" they intend to pay (C.R. 601.2b), before figuring that spell's total cost (C.R. 601.2f). In the case of Reaper King, this "nonhybrid equivalent cost" can be 10 or WUBRG, among other possibilities, and in general, it's this cost that cost reduction effects reduce.