The imperative form is “ne t’inquiète pas”, no “s” needed, that would be an error. I think you are confusing the imperative and indicative moods, which in French follow different conjugation rules.

Present imperative – 2nd person singular: no “s” with regular verbs ending with “-er”

Ne t’inquiète pas

Present indicative – 2nd person singular: “s” with regular verbs ending with “-er”

Tu ne t’inquiètes pas

When you use the expression (ne) t’inquiète (pas) (no ‘s’ here), the used mood is called impératif in French :

• inquiète-toi
• inquiétons-nous
• inquiétez-vous

In this case, we don’t use the mood indicatif (as you said):

• je m’inquiète
• tu t’inquiètes
• il s’inquiète
• nous nous inquiétons
• vous vous inquiétez
• ils s’inquiètent

So here we use the negation of inquiète-toi which is ne t’inquiète pas and became t’inquiète.

Good answers already concerning the mood, which was imperative and not indicative. No need to expand for too long, the S is dropped from the second singular person final for the verbs in -ER in this mood.

I’ll concentrate on your question about the accents varying for every person but nous and vous in the indicative present. It has to do with the pronunciation. When the T-sound is the last one in a conjugation of this verb (we except the E caduc), the É-sound¹ preceeding it turns into an È-sound², and the accent is changed accordingly. French in general doesn’t have very many cases (if any) of words ending with the É-sound + a consonnant³.

• Inquiéter is pronounced [ɛ̃.kje.te], and the T-sound doesn’t complete the enunciation of the word.

• Je m’inquiète is pronounced [ʒə.mɛ̃.kjɛt] (also [ʒə.mɛ̃.kjɛtə] in some areas or circumstances), and the last sound of the word is a T (or a caduc E).

Though there are variations in the way É and È are pronounced, this verb follows the general trend.

A few of the other verbs behaving the same way, with various consonants following the É: abréger (soft G), céder, légiférer, léguer (hard G).

¹ [e] in the international phonetic alphabet.

² [ɛ] in the international phonetic alphabet.

³ _{Possible exceptions:}

• _{Words borrowed from English: email [iːmel] (the neologism courriel is however consistently gaining popularity), freight and brace are commonly borrowed as is in Quebec, but I don’t think they are common in France.}

• _{Some words pronounced in a very familiar tone in some areas of Quebec, perhaps also elsewhere: père, mère, frère, bière can become [peʁ], [meʁ], [fʁeʁ], [bjeʁ] when spoken out loud.}