Siempre llevo un analgésico en la mochila, por si me duele la cabeza en clase.

Llevar means to carry / to have with you. In this sentence, the speaker is talking about something they usually carry in their backpack.

• Siempre llevo un analgésico... = I always carry a painkiller...
• Tengo just means I have
• Traigo means I bring, usually toward the place where the speaker is or toward the listener

So llevo is the most natural verb because the idea is carrying it around.

It describes a habit.

In Spanish, the present tense is often used for things you usually do:

• Siempre llevo... = I always carry...
• Siempre desayuno a las siete = I always have breakfast at seven

So this does not just mean what the speaker is doing at this exact moment. It means this is their normal routine.

Analgésico means painkiller / analgesic. It is more specific than pastilla.

• un analgésico = a painkiller
• una pastilla = a pill/tablet
• medicina = medicine, more general

So an analgésico might be a pastilla, but the words do not mean exactly the same thing.

It is un analgésico because analgésico is a masculine noun.

Both are possible.

• en la mochila = in the backpack
• en mi mochila = in my backpack

Spanish often leaves out possessives when the owner is obvious from context. Here, it is easy to understand that it is the speaker’s backpack.

That said, en mi mochila would also sound perfectly natural. Using la mochila can feel a little more neutral or less emphatic.

Por si means in case. It expresses a precaution.

• Llevo un analgésico por si me duele la cabeza
  = I carry a painkiller in case I get a headache

This is different from si on its own, which means if.

• si me duele la cabeza = if my head hurts / if I get a headache

Using por si shows that the speaker is preparing before anything happens.

Because doler works like gustar.

In Spanish, the thing that hurts is the grammatical subject:

• Me duele la cabeza
  literally: The head hurts to me

So:

• me = to me
• duele = hurts
• la cabeza = the head

This is the normal Spanish structure.
Mi cabeza duele is not natural Spanish.

A useful comparison:

• Me duele la cabeza = My head hurts / I have a headache
• Me duelen los pies = My feet hurt

Notice that the verb changes depending on whether the thing hurting is singular or plural.

With body parts, Spanish usually prefers:

• an indirect object pronoun: me, te, le...
• plus the definite article: la, el, los, las

So Spanish says:

• me duele la cabeza
• me lavo las manos
• se rompió el brazo

rather than using the possessive every time.

Since me already tells you whose head it is, mi is usually unnecessary.
Using mi cabeza is possible, but it often sounds more emphatic or contrastive.

Because after por si, Spanish normally uses the indicative, not the subjunctive.

So the standard pattern is:

• por si llueve = in case it rains
• por si llega tarde = in case he/she arrives late
• por si me duele la cabeza = in case my head hurts / I get a headache

For English speakers, this can feel surprising, because in case often sounds hypothetical. But in Spanish, por si is usually followed by the indicative because it refers to a real possibility you are preparing for.

En clase is a very common expression meaning in class / during class.

So:

• en clase = in class, while class is happening

If you say en la clase, it usually sounds more like you mean a specific class or specific classroom.

Compare:

• No hables en clase = Don’t talk in class
• En la clase de historia aprendimos eso = In history class we learned that
• Dejé el libro en la clase = I left the book in the classroom / in the class

So in your sentence, en clase is the natural general expression.

Yes. Both are correct, but they are used a little differently.

• Me duele la cabeza = My head hurts / I have a headache
• Tengo dolor de cabeza = I have a headache

Me duele la cabeza often sounds a bit more everyday and direct.
Tengo dolor de cabeza can sound slightly more formal or descriptive.

In this sentence, me duele la cabeza is very natural and conversational.

The comma separates the main statement from the reason/precaution phrase.

• Siempre llevo un analgésico en la mochila = main idea
• por si me duele la cabeza en clase = extra information explaining why

In writing, that comma is very natural because por si... is being added as an explanatory phrase. Even if a comma is sometimes omitted in informal writing, using it here is clear and correct.