DOES MICROSOFT 365 F1 HAVE ACTIVE MAILBOX THAT USERS CAN USE TO SEND EMAILS?

No, M365 F1 licenses do not include Exchange Online, as detailed for example here: https://learn.microsoft.com/en-us/office365/servicedescriptions/office-365-platform-service-description/office-365-plan-options#service-availability-within-each-microsoft-365-and-office-365-plan

M365 F3 does.