Without going into rigorous formulations in the language of the logic discipline: outside the mathematics discipline, we can think of an axiom as something that is accepted as fact without rigorous proof, usually because it appears self-evident, being observed countless times, in many different historical, social, cultural etc contexts. What is the difference from an assumption?
Well, by uttering the word "assumption", one already acknowledges that its generality is much less than that of an axiom. It may be an assumption that appears to hold widely, but not with the universality that an axiom carries. So changing assumptions is "fair game" -we try out different assumptions, to see what happens. On the other hand, ignoring an axiom, although certainly not "forbidden", casts a shadow of doubt on any results that will be obtained thus.
Under this view, three out of four of what your professor stated as "axioms", can indeed be seen as such, in my opinion and in the loose sense of the word we are applying here.
SCARCITY: The economics discipline would not exist without Scarcity: both efficiency and distributional issues would be non-existent in the long-run, in the absence of scarcity.
PURPOSIVE BEHAVIOR: In such abstract formulation, I could make it a theorem stemming from the axiom of Scarcity and the biological facts about the human body. But it is more convenient to treat it as an "axiom" (again, in the rather loose sense we are using the word here). Note that I avoided the word "rationality" because it has also a strict use in economics and then it means something different all together (we call preferences "rational" if they are complete and transitive, that's all -nothing to do with purposive behavior).
EQUILIBRIUM Why an axiom? Because it essentially reflects survival (biological or other), in an almost analytical sense (i.e. devoid of meaning). I understand "equilibrium" as a tendency, not as an always-holding situation. Also, an older answer of mine applies.
But in no way should we treat "stable preferences" as an "axiom"- on the contrary, it is an observed fact that preferences can be state-dependent, but also and more widely observed, time-varying (are your preferences the same when young, compared to when middle-aged? I don't think so). And there is a lot of economic research going on with non-stable preferences (the simplest I can think of is a time-varying pure rate of time-preference in growth models). Here we have a clear case of an "assumption".