How often do they happen?
How do they happen?
Do they appear randomly?

there are several types of planetary events...

automatic random events are events that occur automatically at random (i've noticed it happening roughly around ship upkeep time)
anaylzable random events are when you get a msg saying a planet has something to analyze.. this also occurs randomly (possibly as a subset of random events, probably also occurs near ship upkeep time.. haven't had one of these yet)

then there's also these types....
colonization events.... random chance of happening on colonization, always a beneficial effect
invasion events.... random chance of happening on invasion, sometimes beneficial sometimes detrimental... detrimental effects i've seen so far include a random structure is destroyed (mission artis not immune.. can't wait to hear someone cry about losing an omnibase)... -50% defense for a week... or +100% building cost for a week

ive gotten two on my planets, one was a cloak, I can't be bothered going on GL yet, but it increased that planets cloak by +50000.
So it couldnt be scanned out what so ever, and it lasted a week. The other was when I sucessfully invaded, The cost of structure building was reduce by 50% which is quite decent...