You can apply. Remember that the arrest record will be requested along with the disposition. They need to be certified copies. I would run them by an attorney experienced in removal proceedings and criminal issues in immigration.
Make sure you have certified copies of your arrest and disposition. You will need them in regards to future immigration applications. Immigration will not see expunging as an attempt to conceal the arrest and conviction as long as you always check "yes"to the questions of whether you have been arrested or convicted for an offense. Good Luck.
You need to reopen your case and get the copies. If you were convicted or adjudication was withed you need to talk to an immigration attorney as soon as possible. The Government uses those same certified copies to sustain any charges in removal (deportation) proceedings.
Petty theft falls under the petty offence exception of the INA as long as: sentence for the crime may not be more than 1 year and incarceration may not be more than six months. Your attorney (not you) needs to look at the statute under which you were convicted to determine your situation. Do NOT travel without talking to a lawyer.
If you leave as a resident you will lose that status. This means that if you want to return to the U.S. you will have to get a tourist visa. If you are a citizen you will be able to return whenever you want, even if you do not do so for many years. The tax issue is something you will need to discuss with a tax attorney.
This is a concept that works after you have presented yourself as a non immigrant for admission. From the information you provided this period has run out and there is no fraud presumption in regards of your wife's intent when she entered the U.S. Best thing you can do is document your relationship. Good luck.