Hi ​@christoffer.trautmann 

Please find the reasonable feedback for all your pointers,

• When you do this, and “Grant Access”, is that connection made with the account making the connection? Or is it granted “as the organisation”?

The connection is made with the individual account making the connection, not as the organization

• What permission does the connection have, i.e which documents will they be able to embed? Is this based on the person editing the sharepoint site, or the person making the connection?

The permissions for embedding documents are based on the person editing the SharePoint site. They will be able to embed Lucidchart documents that they have access to in their Lucidchart account

• Does this connection apply to all Sharepoint sites, or do you have to repeat it for each new site that wants to use this feature?

The connection typically applies to the entire SharePoint environment, not just individual sites. Once the integration is set up, it should be available across all SharePoint sites within your organization

To learn more, please make use of the help article.

Integrate SharePoint with Lucid – Lucid

Microsoft SharePoint

Hope this helps - Happy to help further!!
Thank you very much and have a great one!
Warm regards