No, that doesn't sound good. As "chodzić" = to go, to walk; and "iść" = to be going, to be walking, "idźmy do kina" would sound like "Let's be going to the cinema"?
That doesn't sound like the best explanation. And I can sure imagine "idźmy" in some other contexts. But not here.
It should have been. Unfortunately, every day I read several comments about a correct answer being rejected. I'm afraid Duo has a lot to work on in terms of bugs.
Let's go to cinema? I'd certainly say it but I'm not native. What do you think?,
It must be a regional thing, then.
In the U.K. (at least everywhere I've been), I've never heard a "show" used for a film/movie. If you go to a "show" it's usually some sort of live performance like a kids' pantomime, a variety act, a cabaret, a stand-up comedian, etc :)