You can start the green card process and still renew your H-1B visa without any problem. An H-1B visa is a dual intent nonimmigrant (temporary) visa, meaning that you can continue to work with an H-1B visa even if you plan to immigrate permanently to the US. Most other nonimmigrant (temporary) visas require that you intend to leave the US at the end of your stay.
In terms of getting married, you should simply be married by the time you file the I-485 (adjustment application).
Good luck!
Attorney Edward Weiner
Boston, MA
http://www.immigration-aux-usa.com/