Are both income statement and balance sheet accounts closed at the end of a fiscal year?

No.

Income Statement accounts are closed at the end of the year, Balance Sheets accounts are NEVER closed, although you do total it up to use as the opening balance next year.

You close your nominal (temporary) accounts at the end of each fiscal year of your company, your nominal accounts consist of revenues, expenses, and dividends. At the end of each year you will close these accounts out to the retained earnings of the company. That will bring the balance in each of these nominal accounts to zero.