Based off info from the UK forums of this game.
Conventions appearances are entirely random. Best way to encounter one seems to be logged in a lot to see when one pops up.
Your chances of encountering one increases with each new area you unlock as they each can have a convention.
Completing as many show missions during the convention is dependent on your boosters or how many diamonds you are willing to spend.
If the convention is successful, I think 2 players. 1 with most shows and 1 with the best show gets a rare item you can only win for ranking first in the convention.
I'm not sure if every convention requires a minimum to succeed, but generally from US1's small player base, the conventions usually fail. I've only been in 3 that have succeeded and that is because a couple of people spent a buttload of diamonds to insta-complete a lot of shows.
If your map missions icon is flashing without an exclamation point, it may be trying to alert you of an on-going convention that has started.
This weeks Temple mission seems like an impossible task for the majority of studios as it would require either more conventions spawning during this week or some players really wanting to dump diamonds to complete enough shows.
Hilariously, we may get a Temple ranking of only a few studios this week, meaning that the rewards for ranking in the Temple ranking might just be moot, as the rewards for the last couple of weeks have been scaled based on the number of studios that met the minimum to rank in the Temple Rankings.