Learn Hardcore Portuguese (Portugal): O professor não quer criticar o Pedro em público. - The teacher does not want to criticize Pedro in public.

If you drop the article and just say Professor não quer…, it sounds incomplete or like a headline, not a normal sentence.
You can drop the article when professor is used after a verb like ser to say someone’s job:

O Pedro é professor. = Pedro is a teacher. (no article before professor here)

Why is there an o before Pedro? Why o Pedro instead of just Pedro?

In European Portuguese it’s very common to use the definite article before people’s first names:

O Pedro, a Maria, o João

So o Pedro here just means Pedro; the o doesn’t change the meaning in English.
Leaving it out – Pedro instead of o Pedro – is possible, but in European Portuguese it often sounds a bit more formal, written, or emphatic.
In Brazilian Portuguese, using the article with names is less common and more regional.

Could I say O professor não quer criticar Pedro em público without the article before Pedro?

Yes, it’s grammatically correct: O professor não quer criticar Pedro em público.
In European Portuguese, though, criticar o Pedro is more natural in everyday speech.
Using or omitting the article with names in European Portuguese is mostly about style and register, not right vs. wrong.

Why is não before quer and not before criticar?

Não normally comes right before the conjugated verb of the sentence:

O professor não quer criticar o Pedro…

Here, quer is conjugated (3rd person singular), and criticar is an infinitive.
Putting não before criticar – quer não criticar – is only used in special emphatic contexts and sounds unusual in this sentence. The neutral, standard order is não quer criticar.

Why is criticar in the infinitive? Why not a second conjugated verb?

Portuguese uses a structure [verb like querer] + infinitive, similar to English want to + verb:

quer criticar = wants to criticize
quer falar = wants to speak
quer sair = wants to go out

So you conjugate querer (quer) and leave the second verb (criticar) in the infinitive.
You would not conjugate both verbs; não quer critica would be wrong here.

Why is there no preposition before o Pedro after criticar?

In Portuguese, criticar is a direct-transitive verb: you criticize someone/something directly, without a preposition:

criticar o Pedro = to criticize Pedro
criticar o governo = to criticize the government

Unlike Spanish (criticar a Pedro), Portuguese does not use a here.
A preposition would sound wrong: ✗ criticar ao Pedro is incorrect.

What exactly does em público mean? Is it like “in front of the public” or just “in public”?

Em público is an idiomatic expression that means in public, in front of other people, not necessarily an actual audience.
It’s about the action being visible to others, as opposed to in private.
If you said no público, that would literally be in the audience (inside the group of spectators), which has a different meaning.

Can I say publicamente instead of em público? Are they the same?

You can say:

O professor não quer criticar o Pedro publicamente.

This is correct and close in meaning.
Em público is a very common, neutral expression.
Publicamente feels a bit more formal or written, and can sometimes sound a little stronger or more general (not necessarily just “in front of a specific group of people” but “in a public way”).

Is the word order fixed? Can I move em público to another position?

You have some flexibility. All of these are possible, with small differences in emphasis:

O professor não quer criticar o Pedro em público. (most neutral)
O professor não quer, em público, criticar o Pedro. (emphasis on in public)
Em público, o professor não quer criticar o Pedro. (puts “in public” at the start for emphasis)

The version in your sentence is the most typical everyday order.

Could I use a pronoun instead of repeating o Pedro, like “criticize him”?

Yes. With an object pronoun (o, a, etc.), in European Portuguese after não you normally put the pronoun before the verb:

O professor não o quer criticar em público. = The teacher doesn’t want to criticize him in public.

If there were no não, both of these would be possible:

O professor quer criticá-lo em público.
O professor o quer criticar em público.

In European Portuguese, after não the usual choice is não o quer criticar, not não quer criticá-lo.

Why don’t we say Ele, o professor, não quer…? Is it normal to omit ele?

Yes, it’s completely normal to omit subject pronouns like ele, ela in Portuguese when the verb ending already shows who the subject is.
Here, quer (3rd person singular) plus the noun phrase o professor is enough:

O professor não quer criticar o Pedro…

Adding ele – Ele não quer criticar o Pedro em público – is also correct, but now ele usually refers to someone already known in the context, and o professor might then be extra information or an apposition.
For a simple, neutral sentence, just O professor não quer… is the most natural.