I have an account which uses SMTP through Gmail, without any problems. This uses port 25 and server smtp.gmail.com with STARTTLS enabled. The only difference perhaps is it’s set up using an app password rather than the standard account password, so it may be worth trying this?
There’s a guide in our knowledgebase on how to set this up: Using a 2-step verification Google Account with QuickFile
Brilliant. You can go to Account Settings >> All Settings >> Sent Email Log for a record of emails sent. These are kept for 6 months.