O professor de que eu mais gosto vai dar aula no segundo semestre.

Because the verb gostar normally requires the preposition de: gostar de alguém / de alguma coisa.

So in the relative clause, that de has to stay:

• O professor de que eu mais gosto
• literally: the professor of whom/that I like most

English often drops or hides that kind of preposition, but Portuguese usually keeps it because it belongs to the verb.

In everyday spoken Brazilian Portuguese, many people do say que eu mais gosto and leave out the de. It is very common in speech.

However, in more careful or standard grammar, de should be there because of gostar de. So:

• Standard / more formal: O professor de que eu mais gosto
• Very common in speech: O professor que eu mais gosto

If you are learning for exams, writing, or formal situations, keep de.

Yes. Since professor refers to a person, de quem is also correct:

• O professor de quem eu mais gosto
• O professor de que eu mais gosto

Both are acceptable. For people, many learners will hear de quem quite often and it may sound a little more natural to some speakers. De que is also fully correct.

Here mais means most, not just more.

The structure de que eu mais gosto means:

• the one that I like the most
• the one I like best

So mais is modifying the idea of liking within that group of professors.

Compare:

• o professor de que eu mais gosto = the professor I like most
• eu gosto mais dele = I like him more

In your sentence, mais helps create the superlative idea: out of all the professors, this is the one you like the most.

Yes, it can be omitted because Portuguese verbs often show the subject clearly.

So both are possible:

• O professor de que eu mais gosto
• O professor de que mais gosto

Both mean the same thing. Including eu can add clarity or emphasis, but it is not required.

Portuguese usually uses the definite article more often than English.

So when you mean a specific professor, o professor is natural:

• O professor de que eu mais gosto... = The professor I like most...

If you said just professor, it would usually not fit this sentence well. The article helps identify the noun as a specific person.

Dar aula is a very common expression meaning to teach or more literally to give class/classes.

So:

• vai dar aula = is going to teach
• depending on context, also will be teaching

It is often more idiomatic in Portuguese than using ensinar.

Compare:

• Ela vai dar aula na universidade. = She is going to teach at the university.
• Ensinar is also a real verb, but dar aula is extremely common when talking about teaching classes.

Vai dar aula is the periphrastic future: ir + infinitive. In Brazilian Portuguese, this is very common and natural for talking about the future.

So:

• vai dar aula = is going to teach / will teach
• dará aula = a simple future form, also correct, but often more formal or less common in everyday speech

In conversation, Brazilians often prefer vai + infinitive.

Because dar aula works as a fixed expression.

So Portuguese says:

• dar aula
• not usually dar uma aula in this general sense

If you say dar uma aula, it usually sounds like give one class/lesson, referring to a single class. But here the meaning is general: he will teach.

Compare:

• Ele vai dar aula no segundo semestre. = He’s going to teach in the second semester.
• Ele vai dar uma aula amanhã. = He’s going to give one class tomorrow.

No is a contraction of em + o:

• em o → no

So:

• no segundo semestre = in the second semester

Segundo semestre usually means the second half of the academic year or calendar year, depending on context. In a school/university context, it often refers to the later term of the year.

The subject is o professor.

The sentence structure is:

• O professor de que eu mais gosto = the full noun phrase
• vai dar aula no segundo semestre = what that professor is going to do

So eu belongs only inside the relative clause:

• eu mais gosto = I like most

It does not control vai dar aula.

Yes, a little. Some small variations are possible:

• O professor de que eu mais gosto vai dar aula no segundo semestre.
• O professor de quem eu mais gosto vai dar aula no segundo semestre.
• O professor de que mais gosto vai dar aula no segundo semestre.

These all sound natural.

But if you remove the de in careful standard Portuguese, or change the structure too much, it becomes less grammatical in formal usage.

Yes, it is natural and correct.

A Brazilian speaker might also say:

• O professor que eu mais gosto vai dar aula no segundo semestre.
  very common in speech, less formal
• O professor de quem eu mais gosto vai dar aula no segundo semestre.
  also very natural
• O professor de que eu mais gosto vai dar aula no segundo semestre.
  correct and standard

So the version you have is a good model, especially for solid grammar.