You can edit the ControlTemplate of the RadDiagramToolbox and remove both elements you've indicated. However, there is one other approach that you can use in case you don't want to edit the entire ControlTemplate of the toolbox. The button right next to the 'ViewGalleries' text has a predefined style which you can control through the RadDiagramToolbox.OpenCloseButtonStyle property. This is why you can create a custom style for that button that sets the Visibility of the button to Collapsed.
Next, the 'ViewGalleries' text is displayed in a TextBlock element of the RadDiagramToolbox.ControlTemplate, which Text value is controlled through a LocalizaitonManager. This is why you can create a custom LocalizationManager class that defines an empty string for the Text value of the TextBlock in question.
I attached a sample solution to demonstrate this approach. Please have a look at it and let me know if it helps.