With so many and such varied questions each requiring detailed replies, your best option is to directly contact an immigration attorney for a personal consultation since this site is only designed for very general, free guidance.
This advice does not form an attorney-client relationship and is merely informative. It should not by itself be relied upon to address a legal concern.
Please note that once someone obtains a green card, they no longer need a work permit. The green card itself gives the holder legal authorization to work in the U.S. That being said, there are other issues present in your case. You need to seek legal counsel before you and your husband travel outside the U.S.
This answer is provided for informational purposes only; it does not create an attorney-client relationship or otherwise require further consultation.
If your husband leaves the country he will be found inadmissible upon return. Even though he got a deferred adjudication on the criminal case it does effect his immigration status. This is a very sensitive matter and I would definitely consult with an immigration lawyer who knows both criminal law and immigration law to get the best advice. I would not leave the country without speaking to a lawyer.