Also, suppose the transaction is not urgent (say changing BTC between own wallets or sending to a relative), can it be sent with very low fee? What's the worse that can happen - will the transcation get cancelled/not go through on the blockchain?
Not directly related, but you mention not urgent transaction. I noticed that usually mempool gets fuller towards end of the week and empty over weekends. So cheapest day to send a transaction is Sunday. Or the fastest day to send cheap transaction.