Yes, to all your questions.
In general, holidays are marked on the calendar; vacations are given by your company. Vacations can occur during holidays; i.e., Christmas vacation. The term holiday can be used to refer to one's vacation; i.e., I'm on holiday next week. Vacation usually means to go somewhere; i.e. I'm vacationing in Miami this winter; Cf. I'm going on holiday to Miami this year.
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