I'm thinking of starting a small web design company and I want to use the word "Apollo" within it. Is this a word I have to ask permission to use from NASA or something? Especially if I call it Apollo 12. Can I use the phrase "Apollo 12" as my company name and be safe?
Apollo is the name of a Greek god and is not owned by NASA. As an aside, I've used the name of an ancient Greek god in my product name, although I used Eos, not Apollo.
"Apollo 12" is different, as that was an actual NASA mission and it is reasonable for them to argue that people would think of the mission when they come across the phrase "Apollo 12".
Even if NASA never noticed you or if you did get permission from them, your company name would be lost in search engine result lists. As cool as the name might sound there will be so many other websites and web pages with that phrase.