"Some colleges" ?? Which ones are those? Teachers have considerable flexibility in how they measure performance in a class, all falling under the heading of academic freedom. I've worked as a teacher in one university and three community colleges; none of them had an strictures on how testing was to be performed.
If said professor collects and grades the homework, by and large the students will do it.
I can't remember the name of the college he said.
This sounds more like an opinion than a fact. Grade inflation is the result of assigning passing grades to students who don't show mastery of the material.
Not. I taught math at the college level (in a community college) for 18 years. I would sometimes give a list of problems similar to the ones I put on the exam. As I said before, a set of practice problems conveys to the students some sense of what topics are high priority and what topics aren't.