I want to marry my fiancee. She is from Greenland here on a tourist visa.

Yes, it is possible to marry and petition for her without having to leave the country at all. However this situation does raise some red flags insofar as she arrived on a tourist visa and marries so quickly. You and your fiancee should consult with an immigration attorney(s) before submitting any paperwork.

Yes, she may be able to adjust status to a permanent resident based on her marriage to a US citizen. But my colleague is correct that marrying in the US on a tourist visa will raise "red flags" because tourist visas are designed for persons who enter without the intent to immigrate. You should consult with an immigration attorney who can learn more about your situation in private and provide guidance. Congrats on your engagement!

Yes she can. But be prepared to prove that your marriage is bonafide as it will be subject to scrutiny.

I agree with my colleague. It is possible for you to marry and for your fiance to stay in the U.S. during the adjustment (green card) process but I would recommend consulting with an attorney to discuss your specific situation before you marry because there are many issues that could arise and complicate the process and each case is different. Best of luck to you both and Congratulations!