How can F1 int'l student in US travel to Canada/Mexico? With what papers?

My US visa is expired but I am legal for the duration of the I-20 (Russian citizen). If I go to Canada, what papers would they check on the border to enter Canada (maybe only DL? don't they only check DL for US citizens, how would they know if I am a foreigner?)?

Would I have to apply at a US diplomatic mission in Canada for a new visa to come back to the US?

I want to go to Canada to practice French but not sure if I want to wait for (if they issue it) a new visa to come back.

You can travel to Canada, Mexico and certain Caribbean Islands on an expired visa as long as you still have a valid SEVIS generated I-20 form. BUT YOU CAN NOT BE OUT OF THE US FORM MORE THEN 30 DAYS!

To come back in the US you will need a valid passport, your expired visa, your I-94 form and the valid I-20 (make sure the school updates the form in SEVIS). Normally, when you depart the US you surrender the I-94 form ,however, in the case of short term travel to Canada & Mexico DO NOT surrender it. If you surrender the I-94 at the Canadian border you will not be allowed back in.

While you could apply for a new F-1 visa while in Canada, if you get denied (which is very likely) you will not be allowed back in the US. As long as you don't stay more then 30 days and don't apply for a new visa you'll have no problem.

Also beware that as a Russian you'll need a visa to get into Canada. If you are very confident that you have a perfect American accent, look american and have an american name they might let you slide by with your driver license but it's highly unlikely (maybe possible if you drive to Canada but there is no way you can get by if you fly in).
Also, only a driver license accompanied by a US birth certificate is an acceptable substitute for a US passport. A driver license alone is not acceptable proof to enter Canada.