Using a lawyer cannot guarantee that the process will be faster. However, there is a better chance that the petition will be complete, showing that the couple fulfills K-1 requirements, and the petition is prepared according to USCIS preferences. This will facilitate the review and processing of the petition which will improve the chances of it being processed efficiently. Talk to an immigration attorney who can review your case and give you feedback.
He'd have to apply for naturalization. Here are the eligibility requirements: http://www.uscis.gov/portal/site/uscis/menuitem.eb1d4c2a3e5b9ac89243c6a7543f6d1a/?vgnextoid=a0ffa3ac86aa3210VgnVCM100000b92ca60aRCRD&vgnextchannel=a0ffa3ac86aa3210VgnVCM100000b92ca60aRCRD
Filing tips on how to organize your application can be found here:
Talk to an attorney to maximize your chances for success.
Here's what the USCIS says:
USCIS has automatically extended the validity of employment authorization documents issued under the last extension/designation of TPS Haiti for an additional 6 months, through July 22, 2013. If you have a TPS Haiti EAD with an original expiration date of January 22, 2012, you are covered by this automatic extension and may continue to work.
To continue to work, you may show the following documentation to your employer and government agencies: