Learn Farsi: او باید دوره را طی کرده باشد.

Actually, no! While you are learning to use past participles to link actions together, here the past participle (کرده) is teaming up with باشد to form a specific tense: the past subjunctive. We use this because باید (must) forces the verb into the subjunctive. Since we are making a deduction about something that already happened ("must have completed"), we use the past subjunctive form rather than the present.

How does the compound verb طی کردن translate to "completed"?

The compound verb طی کردن (tey kardan) literally means "to traverse", "to pass", or "to cover a distance". When it is paired with a word like دوره (a course, cycle, or time period), it means to go all the way through it. So, "traversing the course" translates naturally to "completing the course" in English.

If I want to say "He must complete the course" (present tense) instead of "must have completed", how would the verb change?

You would drop the past participle structure and just use the regular present subjunctive of کردن. The verb phrase at the end would change from طی کرده باشد to طی کند.

Could we drop the pronoun او at the beginning of this sentence?

Yes, absolutely. The ending of the auxiliary verb باشد (bāshad) already indicates that the subject is third-person singular (he/she/it). Including او just adds a bit of emphasis or helps clarify exactly who you are talking about if it wasn't clear from context.