Marriage based visa consular processing can take a year, perhaps longer. Some applicants may be delayed as a matter of discretion. This sort of trip may put the relationship at risk should she have done something that she was unaware of.
The decision to overstay the student visa may concern some consular officials and immigrant aides more than others.
Before you commit to consular process, I strongly recommend an appointment or teleconference with an experienced immigration attorney. Perhaps, you both will stay until she can complete her processing as a conditional resident.
The above is general information and does not create an attorney client relationship.
If you file all the above forms in the U.S., there is no process to pick up in her home country. The I-485 adjustment of status is done in the U.S. If you want her to consular process for an immigrant visa, then just file the I-130 and continue the process while you're temporarily abroad. You will have to demonstrate at the time she applies for the immigrant visa and submits your affidavit of support that you are still domiciled in the US or have taken concrete steps to become domiciled in the US again.
If you go with the adjustment of status route, she can travel abroad once her I-131 has been approved and she has the advance parole document in hand, assuming that she has not become subject to any bars due to overstaying her visa. She may be better off obtaining her permanent resident status first, prior to traveling abroad, but you may wish to seek a more in-depth consultation with an immigration lawyer to go over such issues.