Well, yeah, you can make whatever name you want but it doesn't mean people will interpret it like that. If I heard 'Writing Impro' I would not think the 'impro' stood for 'improvement.' For the first one, 'Impro Theatre', this is likely to be to do with improvisational theatre, although the more usual shortening is 'improv.' For the two company names, when I hear them, I don't think 'improve', I focus on the 'pro' part (which is a common shortening for 'professional.)

Are you sure you didn't see "Improv Theater" or "Improv Group" - with a "v"? "Improv theater" is short for "improvisational theater." It is theater that doesn't have scripts. The actors make up their lines on stage. Sometimes, they take ideas from the audience (e.g., they'll ask for a place, or a profession) and use that suggestion in their play. Sometimes, they show up and just start acting. Groups of people who do that are called "improv groups." As far as "Writing Impro." In English you can pretty much name anything whatever you want to name it. But native speakers will not automatically make the connection between "impro" and "improvement." The vowel sounds are different.