Yes, month-by-month is the standard lease arrangement. If you have a written lease agreement, it will state that a full month's rent will be due on the first of the month and will cover the entire month. Sometimes apartments will allow you to move in or out on a day other than the first of the month, but you have to set that up beforehand.
It's fair. Put yourself in the landlord's shoes. People plan on moving in on the first of the month.
Why should your landlord eat three weeks of rent just because you want to move out after a week?
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