I met a girl when I was working in Taiwan a few years ago, and now we are ready to tie the knot.
A mutual friend of ours did this by marrying abroad, then getting her here on a CR1 visa. Now I am back in America, and am wondering if I can do the same if I marry her here in the States.

She will come here next month to study in graduate school with me. Her student visa (F1) will be good for about 6 months. My plan is to marry her before her visa expires.

Can I have a Justice of the Peace common law marriage with a foreigner? If so, can she apply for CR1? If she gets approved, will she be able to stay here and change her visa status, or will she have to go home to do so?
Roughly how much would this cost? As we are both students without a full time job, cost is rather important to us.

Let me know if I need to elaborate on anything.
Thanks in advance!