If SpaceX makes it to Mars first can the company name it and claim it or blab it and grab it and own the whole planet?
No. The US and 97 other nations are signatories to the 1967 "Treaty on Principles Governing the Activities of States in the Exploration and Use of Outer Space, including the Moon and Other Celestial Bodies." The treaty explicitly forbids any government from claiming a celestial resource such as the Moon or a planet, since they are the Common heritage of mankind.