Can college professors alter their grading scale to pass students? How often does this happen?

Teaching is hard, and writing exams is hard. Anyone who thinks they can set the exact criteria in advance is probably being arrogant. At Chicago (at least in the 1980s) every single course is graded on a curve so essentially every course had its criteria set by the performance of the students. On the one hand this compensates for mistakes in teaching or in writing an exam, on the other hand your performance is partly determined by how good your classmates happen to be.

One of the professors at MIT I tutored under was very angry that students had demanded (so the Institute regulated) that the percentages given to the various midterms and exams were set out in advance, because before that "innovation" he'd been able to adjust the percentages if he'd written an exam not up to his own standards, something you can often only find out after you've given it.

In the UK, these sorts of practices are banned. Every student is supposed to be marked to criteria set up when the course was approved. To make this work, UK universities require the exam to be written very early in the term, checked by a colleague, and then checked again by an external examiner. Then at the end of the year there is something called an exam board, where a panel decides if something went wrong in a course and moves all the marks for the whole course up or down in response. This is better than marking on a curve because the students can help each other learn without hurting themselves, but in other ways it's worse in that the professors cannot innovate and it can be very difficult to update material.