Are submissions logged in the database?

I don’t think CF7 does it by default but we’ve had a lot of luck using this plugin to store the CF7 submissions – https://www.ads-software.com/plugins/contact-form-7-to-database-extension/

Not by default. You have multiple options:
https://www.ads-software.com/plugins/flamingo/ (by the CF7 author)
https://www.ads-software.com/plugins/contact-form-7-to-database-extension/ (as mentioned above)
https://www.ads-software.com/plugins/form-to-post/ (very useful, but not quite what you’re looking for)