Can I change status by marrying a US citizen?

Unless the USCIS somehow determines that your F1 visa was procured with a fraudulent intent (the fact that you came in as a student but never attended school is concerning), you should be eligible to adjust status to the permanent resident. But in your situation, I strongly urge you to talk with an immigration attorney before you file any documents with the USCIS. This is because, once again, the fact that you came in as F1 student but never went to school may be an issue. It would be a different story if you came in, began your studies, but then your situation changed (for example, you suddenly ran out of money or something happened with your sponsor and you lost the source of funds, so you were forced to quit school, or you got sick things like this happen). But my concern is that you never attended any school, and this issue will come up during the adjustment of status process. So again, talk to an immigration attorney.

Yes you are still eligible for "Adjustment of Status" to Permanent Resident (Greencard). One the questions that may come up is your initial intention to enter the U.S. on F-1. When you entered the U.S. you had to show that you had no intention of "immigrating" to the U.S. and you were coming to the U.S. for the sole purpose of attending school. So you should be prepared to answer any questions the officer may ask you about that at the time of your interview. It's acceptable to change one's intention after being in the U.S., so if you intention at the time of the entry was to come to the U.S. as a student and attend school and study, you should be okay.

You can apply to change but need to have a really good excuse why you never attended class. Otherwise, if it looks like you committed fraud to get to the U.S., you can have major issues getting your green card.