This is a complex topic.
I split my answer into 1) translation of your sampe sentences 2) comments on the meaning and usage of the German terms in question (qualitative differentiation) 3) a suggestion of semantic "slide controls" for the meaning of these terms (quantiative differentiation) 4) notes on terms otherwise used in my translation.
Good German translation for the first sample sentence about the school principal:
Der Schulleiter gab eine Erklärung heraus, dass hetzerische Kommentare gegenüber Mitschülern wegen ihrer Hautfarbe oder sexuellen Orientierung nicht geduldet werden.
Good translation for the second sample sentence about the head of state:
Präsident T macht skandalöse Bemerkungen an die Adresse der EU und scheint sich Russland anzubiedern.
2) Diskussion (qualitative approach)
In both cases you could use hetzerisch. It depends on what you want to say.
hetzerisch: somebody tries to make others feel extremely negative about others, perhaps even act agressively against them; the author (he or she who uses the word hetzerisch to describe he utterances of a third person) finds this unethical.
aufrührerisch: quite similar; however, as opposed to hetzerisch, the author sees the person acting aufrührerisch more part of the action than simply telling others to act. A figure like President T would rather be hetzerisch speaking than aufrührerisch because usually he does not do anything concrete but speaking; e.g. he instigates others to use violence, but he himself does not personally beat somebody. Importantly, aufrührerisch is directed against some sort of an authority.
aufwieglerisch: similar to aufrührerisch, i.e. the person acting or speaking aufwieglerisch is seen as part of the movement; also, whereas aufrührerisch is clearly unethical, aufwieglerisch can be meant, sometimes, not that much negatively. There may be situations where one can believe it being a good thing to aufwiegeln others (e.g. citizens in a municipality to take action for some cause in public interest, against a lazy municipal administration); anyway there is an element of "we from below oppose against them up there".
I used skandalös in the president example although its meaning is a little bit different as it relates to (real or imagined) public outcry over morally wrong behaviour or action. But in a sentence like this, in the context of politics, and with the social setting as it is (US society, democratic order, a president acting in a strange and devisive way) you would rather not use aufrührerisch or aufwieglerisch. Eine Menge zu etwas aufrühren or Leute zu etwas aufwiegeln is typically an activity done by people from below, not by those on the very top of a hierarchy. You could, however, use hetzerisch, because jemanden zu etwas aufhetzen can be used for people from below or from above. Hitler, clearly at the top of a hierarchy, still could aufhetzen his followers, and so can President T. So I easily could have used hetzerisch in the second sample sentence; but I wanted to offer an alternative, so I chose skandalös.
The main difference of skandalös as compared to aufwieglerisch, hetzerisch and aufrührerisch is that the latter three all are used for actions where he who speaks (say, President T) clearly wants people to get excited (and perhaps violent), whereas skandalös is used for cases when he who speaks (say, President T) did not clearly intend people to get excited (and angry) but they anyway do. With skandalös it is expressed whether the author thinks that President T intended to cause public outcry or did not intend it; it simply happened. We saw this hilariously in July 2018 after an infamous press conference in Helsinki. The key figure (President T) caused public outcry, and we suppose that in this case he did not intent to, but we are not sure; we use skandalös to describe the effect of the speech, not the intention.
For your sentence about inflammatory comments it would therefore be important to know what you want to say: Do you want to say that these comments cause public outcry (= skandalös), or that they were meant to instigate hateful feelings and actions against the EU (= hetzerisch).
3) Semantic parameters (quantitative approach)
From the elaborate discussion above I derive the following suggestion of bipolar semantic parameters (or slide controls, if you like the image) to differentiate the terms.
intended effect <-> unintented effect
ethical <-> unethical
from top of hierarchy <-> from bottom of hierarchy
involving direct participation in target activity <-> speaker not involved in target activity
For hetzerisch and aufrührerisch I would quantify the semantic values as follows:
Note that translating "race" to Rasse in German would be everything but a good idea. The concept of Rasse is highly contentious (if not inflammatory). In German we have gotten used over the last 70 years to using it only for animals. Hunderassen and Rinderrassen are common and totally acceptable terms, but we do not use the word race for humans, and particularly not in order to group them into boxes such as "white" and "black" (which is a very odd concept anyway). Speaking of Menschen verschiedener Hautfarbe is accepted. The advantage is that it does not assign anything to anybody. There are uncountable numbers of skin colours (actually as many as there are human beings). Interestingly my own skin tends to become olivish in summer.
Regarding hetzerisch it might be interesting that there is the legal term Volksverhetzung, used in penal law. As English analogies I find: sedition, demagoguery, hate speech, incitement of the masses. Volksverhetzung is defined in § 130 of the penal code (Strafgesetzbuch). If you read the definition in this article of the law, you get a good idea of what usually would be seen as hetzerisch even in everyday language (not Legalese).
Regarding "seems to be making inflammatory comments": one could translate the seems by offenbar, but together with skandalös this would be odd: If it is skandalös, it would be inconsistent to speak of offenbar, because offenbar says that we cannot be sure about it because we do not have sufficient information (although we have enough to be inclined to think what we think), but skandalös refers to Skandal which is public uproar, which by definition is a clearly visible thing.
Further related words that can be used in various contexts: