Incursion architect question

If I kill an architect in an alva mission, can they show up again later? Or do they not show up again until the next temple cycle?

If you mean in the same zone, pretty sure you get 3 different rooms

I mean overall for the same temple cycle.

If I kill trap architect in map 1, could they show up as an option in one of the rooms in map 2?

in my experience, yes, i believe they go back into the pool for a unclaimed room.
However i do not recall ever seeing the same architect from removed room cycle in on the same 3 mission map
The ones already active don't change architects until you complete them