The French Wars of Religion were driven by a combination of intense religious fervor and political considerations. (However, that being said, keep in mind also the degree to which these two factors are not mutually exclusive but in fact could feed off each other.)
This period of crisis came out of a context by which the monarchy was attempting to advance its own power and authority at the expense of the nobility (this is one of the key themes of Early Modern history and a trend that would culminate in the Age of Absolutism). In such a context, especially with the monarchy tied closely to traditional Catholicism, it makes sense that a significant subset of the French nobility would have had pragmatic cause to align with Protestantism as an alternative.
To this, you must also add the impact of rivalries and power struggles between various noble families and power blocks: if your rivals had converted to Protestantism, this might make you all the more staunchly Catholic (and vice versa). Thus, these political and religious factors would have been intertwined, feeding off each other in a way that makes them difficult to fully separate from one another.
As far as Henry IV's stabilization of France is concerned, I think the key moments in this history would have been his conversion to Catholicism in 1593 and the issuing of the Edict of Nantes in 1598. However, even while he brought an end to the French Wars of Religion, I think you can make the case that he did not fully restore political stability: after all, the frequency with which he was targeted for assassination, not to mention his being murdered in 1610, would suggest the opposite, in fact.