Me alegra que estés más tranquilo después del examen.

Me alegra literally works like it makes me happy or it pleases me.

• me = to me
• alegra = makes happy / pleases

So the structure is not the same as English I am happy. Spanish often uses alegrar this way:

• Me alegra + noun / clause
• Me alegra tu mensaje.
• Me alegra que estés bien.

In your sentence, the thing that makes the speaker happy is the whole clause que estés más tranquilo después del examen.

Because the verb agrees with the thing that causes the happiness, not with me.

In Me alegra que estés más tranquilo..., the subject is the whole que clause:

• que estés más tranquilo después del examen

That whole idea is treated as a singular subject, so Spanish uses alegra.

Compare:

• Me alegra tu actitud.
  Your attitude makes me happy.
• Me alegran tus palabras.
  Your words make me happy.

So it is alegra because the subject is singular.

Because me alegra que... normally triggers the subjunctive.

Spanish uses the subjunctive after many expressions of emotion, reaction, or evaluation, such as:

• Me alegra que...
• Me molesta que...
• Es bueno que...
• Es una pena que...

The speaker is reacting emotionally to someone being calmer, so Spanish uses:

• que estés

not

• que estás

This is one of the most common uses of the subjunctive.

Estés is the present subjunctive form of estar for tú.

Here is the present subjunctive of estar:

• yo esté
• tú estés
• él/ella/usted esté
• nosotros/nosotras estemos
• vosotros/vosotras estéis
• ellos/ellas/ustedes estén

So que estés means that you are / that you may be, depending on context.

Because tranquilo here describes a state or condition, not an essential characteristic.

Spanish often uses:

• ser for identity, inherent qualities, definitions
• estar for temporary states, conditions, feelings, or how someone is at a given moment

After an exam, being more calm is understood as a temporary state, so:

• estás más tranquilo

not

• eres más tranquilo

Using ser here would sound like you are talking about the person’s general personality instead of how they feel now.

It means calmer, more relaxed, or less worried.

Literally, it is more calm, but in natural English we often translate it as:

• calmer
• more relaxed
• less stressed

It usually implies a comparison with an earlier state, even if Spanish does not explicitly say than before.

So estés más tranquilo después del examen suggests that after the exam, the person is calmer than they were before or during it.

Because the adjective must agree with the person being described.

If the speaker is talking to a male or to someone referred to with masculine agreement, Spanish uses:

• tranquilo

If speaking to a female, it would be:

• Me alegra que estés más tranquila después del examen.

So the ending changes based on grammatical gender.

Because the expression is después de + noun.

So:

• después de = after
• el examen = the exam

When de is followed by el, they contract:

• de + el = del

So:

• después de el examen → después del examen

This contraction is required in standard Spanish.

Yes, but it depends on what comes next.

Use después de before a noun or infinitive:

• después del examen
• después de comer

But después can stand alone if the rest is already understood:

• Nos vemos después.
  See you later / afterwards.

In your sentence, since examen is a noun, de is necessary.

Yes. That is also correct, and it is very common.

There is a small difference in structure:

• Me alegra que... = It makes me happy that...
• Me alegro de que... = I’m glad that...

Both are natural. In everyday speech, many learners hear me alegro de que very often.

Both patterns take the subjunctive:

• Me alegra que estés...
• Me alegro de que estés...

The logical subject is the whole clause:

• que estés más tranquilo después del examen

That whole idea is what makes the speaker happy.

So if you break it down:

• Me alegra = It makes me happy
• que estés más tranquilo después del examen = that you’re calmer after the exam

Spanish often leaves this kind of subject until after the verb, especially with structures like me gusta, me molesta, me sorprende, me alegra, and similar verbs.

Yes, Spanish allows some flexibility.

The original sentence is the most neutral and natural:

• Me alegra que estés más tranquilo después del examen.

But you could also say:

• Me alegra que después del examen estés más tranquilo.

This puts slightly more focus on después del examen.

Both are grammatical. The original version sounds very natural and straightforward.

It is neutral in tone, but estés tells you it is addressed to tú, so it is informal singular.

If you were speaking formally to usted, it would be:

• Me alegra que esté más tranquilo después del examen.

And if speaking to several people in Spain with vosotros, it would be:

• Me alegra que estéis más tranquilos después del examen.

So the register here is informal singular.