Dear Mr. Davis,

"Dear" is the standard, professional way to open an email to a manager. "Hey" and "Hiya" are too casual for this register.

I am writing to discuss the new marketing plan.

"Discuss" is a transitive verb, meaning it takes a direct object without a preposition. "Talk" and "chat" would require the preposition "about" (e.g., talk about the plan).

Would it be possible to arrange a meeting this week?

"Arrange a meeting" is a strong professional collocation (word pairing). We don't usually say "make a meeting" or "do a meeting."

Best regards,

"Best regards" is a perfect, polite sign-off for someone you know by name. "Cheers" is too casual, and "Yours faithfully" is strictly used when you don't know the person's name (e.g., starting with "Dear Sir/Madam").