Learn Farsi: ضمن اینکه پروژه‌ها را محول کند، باید به مشغله‌های خود رسیدگی کند. - While delegating the projects, he must attend to his own busy schedule.

The sentence begins with ضمن اینکه (zemen-e inke), a subordinating conjunction meaning "in addition to" or "at the same time as." This conjunction requires the verb in its clause to be in the subjunctive mood, which is why we see محول کند instead of the indicative محول می کند.

Can I use وقتی که (vaghti ke) instead of ضمن اینکه here if they both translate to "while"?

Not without losing the nuance. ضمن اینکه means "while" in the sense of "besides" or "along with doing [X]," highlighting the juggling of two responsibilities. وقتی که just marks time ("at the time that"), which doesn't carry this "clausal addition" meaning.

Why is there a به (be) right before مشغله‌های خود?

The verb رسیدگی کردن (residagi kardan), meaning "to attend to," "to process," or "to take care of," strictly requires the preposition به to introduce what is being attended to. You will always see it paired this way: به [چیزی] رسیدگی کردن.

Why use the reflexive pronoun خود (khod) instead of just the possessive suffix ـَش (-ash) for "his"?

Using خود emphasizes that he must attend to his own busy schedule, contrasting it with the projects he is delegating to other people. While مشغله‌هایش (mashghale-hâ-yash) is grammatically correct, خود makes this contrast much stronger.