Is it fair to punish absent students with a one-question exam?

I think that this is unprofessional, and is leaving the institution open to all sorts of action (up to and including legal proceedings, if it impacts on a student's progression, for example). For example, what if a student is unavoidably absent?

If it's only "worth a point or two", then I don't think it's worth antagonising people in this way. The sarcastic note left on the door simply underlines how poorly thought-out this whole thing was...

Note that at many Universities the "grade distribution" has to be announced in advance, and changing it later in the term needs often to be approved by chair. This closes the door for this type of idea, and opens the ground for appeals by students.

On another hand, while the exams dates for final exams and midterm exams need to be announced in advance, we are often not forced to do the same for work which is worth a small percentage of the mark. Even if I don't use it, I have the flexibility of having few Quizzes during the term, without the date being announced in advance, as long as they are not worth too much and, most importantly, as long as I announce this at the beginning of the semester.

And I had colleagues which announced at the beginning of the semester that there will be two surprise quizzes during the term(of course technically they were not surprises anymore), each questions worth 1-2.5% percent. And the students know in advance that by missing a class they might lose on those percentages...

In my experience, many professors build a little wiggle room into their syllabi for this reason. For instance, professors may assign 10% of the course grade to "class participation". It is then up to them how they assess this. Some take attendance every day; some take attendance only on a few days; some use this sort of "fake quiz" approach.

I know of one professor who included "Quizzes" as a category of points on the syllabus, and specifically announced that these quizzes would be the kind you describe: unannounced, trivially easy quizzes designed solely to check for attendance. She would have such a quiz on days when she noticed low attendance.

I think the degree to which it is ethical or allowable under policy guidelines depends on hwo transparent the process is and how much the grade penalty is. If, for instance, the syllabus allots 10% of the grade for attendance/participation, I think it's totally legitimate for the professor to assess this via trivial pop quizzes, as long as the penalty for missing them all is no more than 10% of the overall grade. Taking the points out of the "exam" portion of the grade would be less defensible, but in practice I don't think it would cause a major stir unless the penalty was large.