Re mixture check:
I guess doing one or not is up to the instructor. However I have seen a lot of examples in both Canada and the US where the "mixture check" involves pulling the mixture knob out until the engine chokes and then ramming it back in. This doesn't prove much is and is hard on the engine.
If you as an instructor are teaching this then you IMO don't really understand how carburators work. So the bottom line from BPF's POV is
1) I think confirming proper operation of the carburator is worthwhile, and
2) If you are going to do a mixture check do it right.