I haven't kept differing species together, so I don't know about mushroom on mushroom conflict, and I've not personally been successful keeping Ricordea in the distant past, when they were all wild collected and unattached to a rock (FL law prohibits collection of rock more than the size of a US quarter, if I recall correctly, so FL corals were/are chiseled off instead of collected on their rock).
Rhodactis did a bunch of damage to a small Acanthastrea I have; that's my only evidence for their aggression. I have to kill mine back occasionally with Aiptasia-X.