How much of us turn off Random Events?

Why would you?

It's pretty much expected in MP that you turn them off...when they aren't turned off, it usually is because the host forgot...

Rarely. Usually when I am testing something for my mod and I don't want its interference.

Rarely...  I think unexpected/random events add to the fun, without shifting the balance drastically.  That said, I do turn them off when playing against a single AI opponent on a tiny map, but I usually play AI games with 4+ players.  MP games are a different story altogether.  In those cases the majority vote counts.

I usually have them on if I host the game, and I've rarely noticed them having a major impact. The most influential ones like economic upturn/downturn tend to affect all players so that doesn't change much. Of course I haven't gotten that mad Vasari random event yet, I suppose that could be a real game shifter if it occurred early enough.

The Mad Vasari, the partisans, and the gamma burst are the worst by far...they can royally screw a player, either by wiping out their HW population or by feeding their opponent XP for their titan/caps...they are fine I suppose for SP (if you like having seizures), but for MP they are shunned...

The first time when you play with these new Random Events you are like a little kid with a new gift, and it's also so shinny - yeah! But when you play a little more you realize that there is nothing fun in: ...

•  Your economy to going up and down, up and down, over and over again. 
•  Chance of luck to determinate your strategy skills - random killing of population, disabling weapons and abilities. 
  
  And so much bugs and problems they added with this last DLC. I'm getting feel more and more that they tricked us. Like a kid with lollipop instant of a new bike.  
  
   

Stardock is also well known for fixing a lot of these issues. So sometimes the lollipop is a placeholder for ice cream.