Google-docs – How to configure Google document so that table of content is automatically updated

Yes, it's possible.

On way is to place text markers as it's done on mail merge scripts.
Another way is to create the table programmatically by using Class Table

I would use the transpose command of Sheets for this.

1. Copy-paste the table into spreadsheet, e.g., into the range A1:E7
2. Elsewhere on that sheet, enter =transpose(A1:E7)
3. Copy-paste the output back into the document, which will recognize it as a table.

I tested this with a table similar to yours, containing linebreaks in some cells. The only thing lost in the process was the text alignment: the output of transpose command always has the default alignment (right-aligned numbers, left-aligned text). But this is not hard to fix manually.