Yes, completing missions for Event_Agents
will incur both positive and negative standings with all Empire Factions. Those missions can only be completed once per character.
The only exception to that are the Epic Arc Agents. They give a large Faction standing increase with no derived standings towards the other Empire Factions. However, a few of their missions will incur some negative standing for ship and or structure kill. That amount is small when compared to the actual amount of positive standing gained for completing the Epic Arc. Also the Epic Arcs can be completed every 3 months.
If there's a problem with your Faction standings, you might want to check out the Faction Standing Repair Plan