The only things I had not set to random were research speed (normal) and difficulty (tough).
So what I got was a medium galaxy with only 2 opponents.
Habitable planetes were extremely rare but anomalies were extremely abundant.
I as Terran started in the middle of the galaxy while my 2 opponents Arceans and those extra evil Drengins started in the corners.
Due to that layout I could add several more surveilance ships. I found so many bc-ressources that I never had economic problems despite having very little tax. Also the tourism helped a lot because I quickly grew influence-wise. (I chose to modify the terrans to boost their advantages with +30% influence and +30% diplomacy.)
A key moment was one of the galactic-council meetings where I could decide that everyone has do give 4 techs by having more than half of the influence already. With that I got the special Arcean-Colonizing tech toxic and could get 2 planets near the drengin which the arceans did not colonize because they were out of range and/or too deep in my influence-area.
Besides me killing of the single minor-race the game actually had no action whatsoever. I was behind in military all the time but the aliens would not attack because of the high diplomacy-bonus I had.
I built influence-improvements and starbases and while only one of the arcean worlds defected that was enough to gain an influence victory without ever fighting a war.
I think the conditions of that game were absolutely perfect to pull that off. But it did not feel like an accomplishment due to the randomness giving me a perfect map for that kind of victory.
Well, I'm excited to see what kind of game the next all-random will be. ^^