I had the same thoughts before I done a few.
Over here I am pressuming is the same everywhere.
You get different sorts of open night mics. You get ones like a folk night where folk can all jam together or it goes round in a circle to have a turn of playing. If others know it they will play along, if not then they keep quiet and you play the song.
Other ones where you have to put your name down and they call you up. It can be for 1,2 3 songs or whatever, depending on how busy it is.
As for do you play your own or do a cover... I would say it depends on if anyone else is doing there own songs or just covers. Covers I would say would be easy to do for your first few to give you the confidence fi someone is lacking any to play in front....the nagain maybe better doing your own since any mistakes will not be picked up so much.
Mistakes!!! dont let any wee mistakes put you off, just carry on. Folk that dont play will not mind unless they are **&^$%%^ 's and any other musicians will have made one or two in their time, they know mistakes happen.
I prefer the folk sessions, I have a better laugh at them as well as good music. I sometimes play a toned down stiff little fingers song so it is not always folk music etc that gets played at these nights.
Whatever you are happy and confident at doing I would say go for both if you get 10 minutes. Do a cover and a song of your own. Cover first to get attention then yer own and see what reaction you get. But a song of your own will not probably get as good a reaction as a cover as folk will not know it. If they do give a really good reaction you know yer on a winner.
good luck and let us know how ye get on if you go ahead.
ye get some that are cut out for the job and others just get by from pretending