Can any Confluence user edit a Smartsheet sheet without having a Smartsheet account?

Yes, if you use the correct URL to share your sheet. 

Use the Edit by Anyone to share and everybody can edit the sheet.

Can I embed sheets with a free account?

Yes, users with a Smartsheet account can embed sheets in all available modes.

Can I embed Smartsheet sheets without a Smartsheet account?

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

The command /smartsheet is triggering a different Smartsheet app.

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