The admiral AI will attack anything marked as Hostile it can defeat. So they sometimes go around, attacking Spaceborne aliens. Might also have been their or your Pirate event.The aforementioned empire has been sending survey ships into my territory. I largely ignored them because I have been busy with building up my colonies and stations. However, just now they sent in a fleet. I sent my fleet to the same place, and they left.
Rivalry will at first give you influence. It can be a major source. They will rival you back.At this point, I can't just do nothing. I already decided to close my borders. Should I also declare them my rival? I don't see a reason why not at this point because dimplomacy will obviously get me nowhere, but I just want to see what you all think.
I now am also recently in contact with a second empire that is much more compatible with mine. I immediately got us both into a NAP and have a trade deal going with them. If I were to go to war, how likely am I to get them to fight with me?
It also has ramifications for Diplomay: "Mutual Rival" and "Rival to Allies" are relevant Diplomacy modifiers.
How likely to get them into War? Just look. Declaring war does nothing until you confirm the Wargoals and you can always invite them.
They will need contact with them and see if they dislike them as well. You can not control either of those.
If you told us the Ethics and AI Personality of the 2rd empire would could help you more.
Thus far we have:
You are stock UNS
They are Authoritarian/Xenophobe/Materialist wich I asume would give them a agressive AI personality. Propably Hegemonic Imperialists.