Is it permitted to borrow a black hat for Davening without permission; does it go under the category of a mitzvah?
Yes, if the assumption is that people wouldn’t mind. My assumption is that nowadays most people would mind. When permitted, see conditions here: https://asktherav.com/article-248-using-anothers-tefilin/ And here: https://asktherav.com/article-617-the-tale-of-the-switched-tallis/ See also…