Yes, for consistency. Test 2-3 with your audience or trusted peers, then commit to one as your official version for at least 6 months.
No. A mission statement explains why you do the work and who it's for, while a tagline is a short, catchy phrase used in marketing. The mission statement is longer and more personal.
A brand message focuses on the audience's problem and outcome for marketing purposes. A mission statement is more about your personal "why" and long-term purpose behind the brand.