1) Can DT Register allow registrants to log in and see what events they are registered for?
Yes, in two ways. One, you can use our Community Builder or JomSocial plugins that show event registration history in the user's profile. Or Two, use our frontend User Panel. When you create a joomla menu item for DT Register, you'll see this User Panel option. This link will show the user's registration history in more detail and also give options for cancellation, editing and additional payments... based on the parameters you have setup.
2) Is there any way on the frontend or backend to display registrants by the organization they belong to?
On the backend, no. On the frontend, if you use the Attendee's list and have the Organization field set to be included in that listing of registrants, a user can sort the list by Organization if they want to. Thanks!