How does a green card via marriage work?

If you do not have a criminal record you can potentially adjust your status without leaving the country. You will have to satisfy the requirements for financial support and that may be difficult or impossible without someone to guarantee support.

You may process your green card in the US since you entered lawfully. Your overstay of the visa will be excused based on your marriage to a US citizen. You will likely need a joint financial sponsor to confirm you will not go on welfare. If you leave an process, you may be subject to bars and then need a waiver. You should definitely process here.

You do not need to go back to your country and you do not need a waiver. She can file for you if you are married. The only issue is I-864 affidavit of support. You should contact a lawyer for advice.

Generally, someone who entered the U.S. lawfully and with inspection (such as entering with an F-1 visa) may adjust status in the U.S. (without needing to go outside the U.S.) as the spouse of a U.S. citizen, notwithstanding that his visa has expired and he has unlawfully remained in the U.S. This is true regardless of whether the U.S. Citizen spouse is unemployed and has received public assistance for herself and/or her child. In cases where the U.S. Citizen spouse has insufficient documentable income to support the foreign national spouse at the required level, however, it will be necessary to have a "Joint Sponsor" for the adjustment of status application. The Joint Sponsor need not be a relative, but generally may be any U.S. Citizen or Permanent Resident with sufficient income who is willing to have the responsibilities specified in the required Affidavit of Support form. There is no legal requirement for a married couple to engage an immigration lawyer to represent them in the adjustment of status process, but especially where the foreign national no longer has a valid nonimmigrant visa it would be wise to engage an immigration attorney. The adjustment of status process, including the various forms that must be filed, may look deceptively straight-forward, but there are many "traps for the unwary," and the consequences for errors may range from substantial delays to denial of applications (for a foreign national who is unlawfully present, a denial is likely to lead to the initiation of removal/deportation proceedings). Beyond errors that the couple may make, it is not uncommon for the USCIS itself to make errors that best can be addressed by an experienced immigration lawyer at the earliest possible time, when that process is least complicated and least expensive.