Once he gets EAD, and travel authorization, can he still work legally in US and make business trips without any issues?

Generally, in a marriage-based adjustment of status case in the Immediate Relative visa category, a foreign national can be granted an EAD and Advance Parole for use while the I-485 application remains pending. The EAD authorizes any lawful employment in the U.S. and the Advance Parole authorizes re-entry into the U.S. after travel outside of the U.S. Once your husband may have the EAD and Advance Parole, he should be able to be employed and to make business trips abroad. Further information is needed in order to advise about the Affidavit of Support required from the Petitioner in this type of case. Generally, if the Petitioner does not have sufficient documentable income to meet the I-864 requirements, the couple should get a joint sponsor. Generally, any U.S. citizen or Permanent Resident who has sufficient documentable income and who is willing to assume the responsibilities associated with an Affidavit of Support will qualify as a joint sponsor. Details about the Affidavit of Support can be more complex than they might appear just from reading the I-864 instructions and reading the statute, and the same applies to other details governing immigration eligibility. It would be wise for you and your husband to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex application process.