Can a 26 year old permanent resident be deported for a misdemeanor crime?

A misdemeanor conviction can trigger serious immigration status consequences, from inability to get a 10-year green card to getting deported. It depends on the statute for violating which your daughter-in-law gets convicted, and on the penalty imposed by the court. If your daughter-in-law did not yet get convicted, she should get a defense attorney and she should make sure that her defense attorney knows exactly what dispositions of the case are unacceptable because of the immigration law consequences. If she already has been convicted, talk to an immigration law attorney to find out what immigration law consequences are likely to flow from the conviction. Sometimes, reopening the criminal case can help to avoid an otherwise inevitable deportation. But reopening a case takes time, and the Immigration Court will not wait for that process to be completed; so, if it has to be done, it's better being done ASAP.

It depends on what the misdemeanor charge was. Some misdemeanors can be considered aggravated felonies for immigration purposes even though they are classified as misdemeanors on the state level and will lead to deportation. Other misdemeanors can be considered crimes involving moral turpitude (CMTs or CIMTs) and will make a person inadmissible from returning to the U.S. should she travel abroad. I suggest that you contact an experienced immigration attorney for a face-to-face consultation and give him/her all of the facts surrounding your daughter-in-law's situation. He/she would then be in a better position to analyze her case and advise her of her options.

Yes, despite her admission as a legal resident alien, if your daughter in law's misdemeanor crime qualifies as a crime involving moral turpitude may be deportable. But despite this, she may nonetheless be eligible to waive her deportation upon showing of hardship to her qualifying US relatives if she is deported from the US. You cannot assume she is in fact eligible for a waiver and so, you must instead consult with an attorney with experience in deportation defense to carefully assess if (a) if her crime indeed qualifies as a deportable offense; (b) whether she is eligible for a waiver and (c) examine the procedure and requirements for obtaining that relief