Do foam details rot
my house ?

Of course they collect moisture and even rot your house. In 
my experience there is water trapped behind the foam, saturating everything behind it.
We just tore off some wood moldings that were covered
with mesh and synthetic "stucco" (sic). The installation was
only three years old, and the wood was soaked with water like a sponge.
It seems the EIFS base and finish absorb water like a thirsty
sponge. The water is then trapped against the wall.
These foam shapes would probably work if they were flashed.
For more about EIFS, click here.

Flashing all the way back to the substrate should prevent water 
infiltration. I think so, anyway.