Is this a correct use of auquel?
C'est le garçon auquel je pense
Is this a correct use of duquel?
C'est le bœuf duquel je parle
Is this a correct use of dont?
Il est le professeur dont je parle
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Yes, this is correct.
This is incorrect. "duquel" may be translated as "from who/which", so you should is it in a sentence like this:
Je parle du pays duquel tu es né
Note that the use of « duquel » is very rare, and seems to be formal. On an oral conversation, I would not use it, I would change the structure of the sentence to avoid it.
The usage of "dont" is good, but the beginning is odd. In french, you don't use "il est" when you are describing a person, you will use « c'est », or « c'est lui »:
C'est le professeur dont je parle
C'est lui le professeur dont je t'ai parlé
Using « c'est lui » emphasis the fact that you are talking about THIS person in particular, and not an other one.