Yes, it is possible for the person to person to be married and adjust status in the United States if the person overstays
his visa and marries a US citizen because an overstay is forgiven based upon a marriage to a US citizen.
The misdemeanor would depend upon the nature and type of criminal activity because one crime involving moral turpitude could result in the deportation
of the person. The general rule is that any crime involving fraud, deceit, or intent to harm others are considered crimes of moral turpitude. Some examples are arson, blackmail, receiving stolen goods, assault, forgery, or fraud. In most cases, DUIs are not generally considered crime involving moral turpitude unless a person is harmed or a third party's property is damaged.
In any event, it is always a good idea for a foreign national to retain an immigration
attorney to review his or her criminal case to avoid pleading guilty to a crime that will have a detrimental impact on whether the person can remain in the United States or be required to leave.
If the criminal defense attorney is able to avoid a conviction then the arrest would not have any impact on a future immigration application.