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.)