Show map notes under map? Is it posible to show the selected data under the map also? When i enable "københavn" as show at the picture, then there should be a list of resturants downunder also?
Yes - this is possible with the great views support in the Leaflet module you are using: 

https://www.drupal.org/node/1645460

I'm assuming that you are using Views to render the above map given the number of data points you are displaying (and if not you should consider it). If so, then the easiest way to do this would be to edit your view, and add an 'attachment' display based on the same arguments that you are using for the above view (including the contextual filters you are using) and then change the display format to whatever you want (ex. an html list, a table, etc.) Attach this to your map view (display below) and you're all set.

Another option would be to do the same but render it as a block display instead and then place this using context or block admin to place this right below your map view.