Can any Confluence user see an Airtable database without having an Airtable account?

Yes. The information provided by Airtable allows the users to see the database without being logged in to Airtable.

Can I embed the database with a free account?

Yes, users with a Airtable account can embed the database in all available modes.

Can I embed the Airtable database without an Airtable account?

Yes, users with no Airtable account can also embed the Airtable database as long as they have the links to the boards.

The command /airtable is triggering a different Airtable app.

It can happen that different apps are called typing the /airtable command if they are installed at the same time in Confluence. Make sure to select Airtable for Confluence from the dropdown menu. Alternatively, you can look for the app using the Macro menu.