So I remember reading somewhere in the Dragonborn expansion that Indoril has ceased to function as a Great House, and that when someone joins the priesthood they have essentially joined Indoril. Does this mean that other Great House members can no longer serve in the priesthood, since they would belong to two houses? Or would this not be the case since Indoril is not a Great House and does not have a place on the Council? And if this is true, can one be a member of Hlaalu and other Houses since it is not a Great House or on the Council either?

I think that being a member of a minor House precludes you being in a Great House in addition - you may be able to put your support behind one of the Great Houses, as most have, but you're still splitting your loyalty between groups.

It would make sense for the Temple to allow members of Great Houses to join their ranks. They want all Dunmer behind their banner in terms of religion, and are no longer functioning mainly as a political faction as Indoril was. Now sure, someone who's splitting their time between serving a Great House and serving the Temple isn't likely to hold a high rank in the Temple Hierarchy, just by virtue of time committed and possible conflicts of interest. However, the idea is that the Dunmer as a people are to return to their pre-Tribunal roots of worshipping the three Daedric Princes they do. That means that the Temple cause and any Dunmer House cause should not be mutually exclusive.