If I make a new account somewhere, if I'm paired waaaay down in a small tournament (has happened twice), or if someone new comes to the chess club and I play them.
I'd say at least a few times a year.
Oh, or if someone wants to play a game on chess.com for fun. I've played a few of those.
Sometimes, standard formula is currently set at -300, 3000. Mostly when I am bored. And mostly I will lose once in a while 
Talking about online chess right now, hmm.
OTB it sometimes happens you get paired down in tournaments. Sometimes.
Almost never. It's hard to find someone ranked at about -150.
Happens often in the first round of an open tournament
Only in tournaments where such players are entitled to play.
When I play 1500 opponents in such tournaments I feel sometimes more exhausted than after a game against a 2400 FM. When we make a post-mortem I learn that most of my calculations were actually useless as the opponent would not even think of the moves which I find as his threats.
I am frequently surprised by lower-rated players' moves, taking some time to realize they simply blundered and not just made a brilliant combination to crush me 
The masters' moves are much easier to understand, they follow well-known principles and tend to calculate the same lines I ponder on
How often do you play players 500 or more points lower rated than you