I don't see the mega-events since I play with them off, but I usually get several of the regular events, influence, economy, pop growth, etc. I play TA at obscene to suicidal on on large to huge maps with 4-7 races. Games usually run around 4 years.
As far as a specific event, I think that's randomized just like topics for UP meetings so an event is pulled out of a hat when one is triggered. Based on what I've seen in the error log, a random event rolls every turn so my guess is there's a big number of "no-event" events in a pool with a small number of "action" events. I often go several games without seeing one event or another. I'd say that's just luck of the draw.
There's one event I really like that almost never comes up. That's the ultra-drive one that increases all ship speeds 50%. I haven't seen that one since Dread Lords. I thought they deleted it, but it's still there because I finally got it in a recent game.
As far as event frequency with respect to difficulty, I haven't played on a lower difficulty for a long time so I don't remember, but I think random events occur at the same frequency regardless.
