Is it possible to select multiple tasks and change their dates all at once?
Yes, that is possible for tasks with non-recurring due dates.
To select multiple tasks, please hold the CTRL key and then click on each task to select it. Once you do, you will see a menu at the top with options that can be performed on all selected tasks.
Sadly, one of those options is not to "*Remove* Due Date" (can only change it)... feature request!
+1 for Remove Date.