Learn Russian: Вряд ли ты сможешь убедить директора, он же эгоист. - It is unlikely you will be able to convince the director, he is an egoist after all.

The particle же adds the flavor of "as you already know" or "after all." By saying он же эгоист, the speaker treats the director's selfishness as an obvious fact that both people are aware of, using it to emphasize why convincing him is hopeless.

Could I use ведь here instead of же?

Yes, you could say ведь он эгоист. Both же and ведь can translate to "after all" or "you know," pointing to a shared fact. The main difference is word order: ведь usually goes at the start of its clause, while же almost always comes immediately after the first word of the clause (here, right after он).

Is вряд ли a single concept, and does it change the sentence structure?

Yes, вряд ли means "unlikely" or "I highly doubt it" and is always written as two separate words with no hyphen. It doesn't force any special grammar changes; you just place it at the beginning of the clause, followed by the normal subject and verb: Вряд ли ты сможешь.

Why does директора end in -а? Isn't it the direct object of the verb?

Yes, it is the direct object receiving the action, so it takes the accusative case. Because a director is a living person, Russian applies the animate accusative rule for masculine nouns, making the accusative ending identical to the genitive ending. If the object were an inanimate thing, it would look just like the dictionary form.