Can I just go to the United States and marry him anyway, and still
live in Canada?
Basically, yes. But, if you tell the border patrol that you are engaged or headed to your wedding when crossing the border to attend your wedding, you may be denied entry, because you no longer qualify for a tourist visa or student visa, and now need a fiancee visa to enter the U.S. This can prevent you from crossing the border for a long time while a financee visa is processed. (I've seen this happen in real life more than once when I lived in Ann Arbor, Michigan which is near the Canadian border.)
Once you are married, it is perfectly legal to live in Canada, although this may lead to tough questions from U.S. or Canadian immigration officials about whether you had a bona fide marriage if you later seek to gain U.S. citizenship, or he seeks to gain Canadian citizenship based upon being married, since not living together is circumstantial evidence that you have a sham marriage.
This said, immigration officials tend to be more suspicious of sham marriages when one spouse from a poor country seeks citizenship in a rich country than they are in U.S.-Canadian marriages in either direction. Immigration officials are more suspicious if you do not have children together and are less suspicious if you have children together.
Do I have to report this to anyone?
You need a marriage license from the state or local government as applicable where the marriage takes place and it must be returned with signatures from the officiant and a couple of witnesses within the time set forth in the license.
If you change your name upon marriage, you need to change it for purposes of all of your identification documents including your passport, usually by submitting your marriage license to the appropriate officials.
If you are asked by a border crossing official if you are married or engaged to be married you are required to answer truthfully, but you do not have to volunteer the information.
If either of you applies for a permanent residence visa or citizenship in the other country the immigration officials of that country must be informed on the application for the visa or citizenship.