Sorry, so with a new mail profile (created in Control Panel -> Mail (32-bit), it still pops up "Enter network password..."
Error 0x800CCC992 Server rejected your login means there is issue with login/password. Go to Tools menu, then Account Settings..., then select your gmail account, then click on Change..., then click on More settings, then make sure the option "User same settings as my incoming mail server", then check.
If needed, make sure Less secure apps option is Allow, then check sending:https://www.google.com/settings/security/lesssecureapps
If it does not work, please go to Start->Control Panel->User Accounts->Manage another account, and create a new account, and select the Administrator option when prompted, then restart the computer, then log in the new account, and then open Outlook and it will prompt you to set up your gmail, then see if sending works fine in the new account.
Please let me know. Thank you.