Learn Hardcore Italian: Il professore dice che è essenziale che ognuno studi la propria materia con pazienza. - The teacher says it is essential that everyone study their own subject patiently.

The first che is the conjunction after dice, introducing the reported clause.
The second che is part of the impersonal expression è essenziale che, and it introduces the clause that requires the subjunctive mood.

Why is the verb studi in the subjunctive (and not studia)?

Because the impersonal expression è essenziale che triggers the subjunctive. In Italian, expressions of necessity, desire, doubt, emotion, etc., followed by che, require the subordinate verb to be in the subjunctive (here: studiare → studi).

Could you explain ognuno? Is it the same as tutti or ognun?

Ognuno is an indefinite pronoun meaning everyone or each person.
It differs from tutti in that ognuno emphasizes individual action (“each one studies”), while tutti emphasizes the group (“all study”).
Ognun is an archaic variant; modern Italian uses ognuno.

Why is it la propria materia and not la sua materia?

After indefinite pronouns like ognuno, Italian prefers the reflexive possessive proprio/a to clearly link the ownership back to the subject.

ogni studente studia la propria materia = each student studies his/her own subject.
Using sua could be ambiguous about whose subject you mean.

What role does con pazienza play here?

Con pazienza is an adverbial phrase (“with patience”) describing the manner in which everyone should study. It modifies studi by specifying how the action is to be carried out.

Can this sentence be rephrased more directly without changing its meaning?

Yes. For example:

Il professore dice: “È essenziale che ognuno studi la propria materia con pazienza.”
This uses direct speech instead of reported speech but conveys the same idea.