As we know, the bounty period is 7 days. After the bounty ends, there is a grace period of 24 hours to manually award the bounty. During all this time, does SO send me a message to remind me to award the bounty? And does SO send me a message when the question have a new answer?
Yes. If you haven't awarded the bounty yet, you get a notification when the grace period starts:
Your bounty on question "How can I improve questions marked as duplicate?" is completed and will be auto-closed after a 24-hour grace period.
You'll always be notified of a new answer to your question, regardless of whether you set a bounty on it – and as long as you haven't disabled the option in your preferences, both the grace period notification and the new answer notification will be emailed to you if you don't read them on-site within a certain time.