Can I still apply for a green card even though my visa has expired?

Your case poses a few factors that should be discussed in detail. Have a consultation with an experienced Immigration Attorney BEFORE you do anything that may jeopardize your case in the future.

Yes, your US Citizen spouse can file the required concurrent petitions on your behalf for family-based adjustment of status to lawful permanent resident even though you are currently an "Overstay" (as long as you entered legally initially and have not departed since the expiration of your student visa). Do not leave the country until you have been granted advance parole authorization as part of the green card filing processes. You should both seek professional guidance from an experienced immigration lawyer to help you with the processes involved since they can be both complicated and lengthy.

From what you have described, yes, you should be able to have your wife file an immigrant visa petition on your behalf and adjust status without leaving the US.

You can apply for a green card through your spouse despite your expired status. I recommend that you talk to an immigration attorney if you are not sure how to go about the application.

Yes, you can apply for adjustment of status to permanent residence on the basis of your marriage and you can remain in the US while that is being processed. You do not need to return to your home country.

You can apply for your green card without having to leave the US. The process is called adjustment of status and takes approximately 3 months from start to finish.