Yes, in American English, you can omit the word "on". (People from other English-speaking countries may have other opinions) Good luck!

Yes. Omit 'On'. Then it sounds like American English. Or you could say.... I'm in a show at my school next Friday. I'd love it if you could come. Or... I'm in a show at my school next Friday. I'd love to see you there.

You can omit it. Better ways to say it: "There's a play at my school next Friday" "There's a play on stage next Friday at my school" We don't use "on" very much for shows in the U.S. but we say "on stage", "running", "showing", "playing".