Yes it is quite possible. The Myth Busters tackled this myth in one of their episodes. Your eye sockets aren't connected to your nose so the power that comes from sneezes won't be transferred to your eye sockets. Anyway, there aren't any muscles behind the eye to violently contract and pop your eyeballs out of your head. So yes, it is possible to sneeze with your eyes open without the fear of them bursting out of your head.
No, it is impossible to sneeze with your eyes open. Researchers discovered that closing your eyes while sneezing is an intuitive reaction triggered by your brain to avoid bacteria and debris from going into your eyes. Sneezing as an act is usually involuntary allowing your body to go through a systematic process. Check out http://health.howstuffworks.com/human-body/systems/eye/sneeze-with-eyes-open.htm to learn more.
Yes. It is possible. Although your eyes naturally close when sneezing,you can train yourself to stop them shutting completely. The sun sometimes causes me to have long sneezing bouts when driving and I can now keep my eyes open and thus avoid running into something
No and yes. No because your heart skips a beat when you sneeze, involuntary movements make your eyes close so you won't be hit with bacteria to close your eyes. Yes, but only because Chuck Norris can :)