could anybody help me out? I want to create a multiple selection box (like you can see when you click on the processing saga tool "Merge vector layers") with the Qt designer.
In QT you may use the QList Widget. There are various selection options under selectionMode setting you may pick from to enable mutli text selection, see graphic below:
For selecting features or all features in a layer you first need to create QGIS Vector layer that references the layer on disk or in a database, and then perform the selection query. There are many posts on this site and others (search for pyqgis create vector layer, pyqgis select layer or select feature).