'Start Date & Time' and 'End Date & Time' has a ellipsis (three dots) button that does nothing when you click on it. Since the 'dtstart' text box is readonly and required, this means I have no way to add a value or an event for that matter.
Registration Open Date and Cut-off Date have a calendar icon instead of an ellipsis button, but they don't work either.
This is not a bug, but would be due to something else in your admin breaking the loading of scripts needed for DT Register. In most cases, this is due to using a poor or outdated text editor. Editors like FCK and CK are known to cause problems are normally found when there are problems like this. Using the default editor or JCE (the best option available), you wouldn't have this type of problem.
With an updated editor or quality, if you still have this type of problem, log a ticket.