It is the difference between the affirmative, the negative and the interrogative forms.
This is the simple present tense and do/does/don't/doesn't are the forms of the auxiliary.
When the auxiliary is present, the main verb reverts to its root form, i.e. without 's'.
We don't use an auxiliary in the affirmative: he works hard (except in the emphatic affirmative: he does work hard!)
In the negative and interrogative, we use the auxiliary for the third person singular (he, she, it) and the main verb reverts to the root form. Let's try the verb to want instead.
Affirmative: He wants to write a report. No does, so wants has an 's'.
Negative: He doesn't want to write a report. Doesn't is present so want goes to the root form.
Interrogative: Does he want to write a report? Does makes want revert.
So your sentence "she doesn't have to wear a uniform" is correct.
As for your other sentence: "people need to realise that their life doesn't revolve around them only", this presents other problems¹ but as far as "doesn't revolve" is concerned, that is correct.
¹Maybe you should ask another question.