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.
You made me curious about transaction fee in different days of week.
I calculated the average transaction fee for different days of the week in the past three months. (Raw data from
https://bitinfocharts.com/)
Monday: 0.70 $
Tuesday: 0.70 $
Wednesday: 0.71 $
Thursday: 0.71 $
Friday: 0.74 $
Saturday: 0.63 $
Sunday: 0.57 $
You were right. I hadn't ever noticed this.