First question is regarding when the tasks appear in my Thunderbird/Lightning calendar , but not on my Android calendar. When I look at the associated Tasks app on Android, I don’t see a list including due dates- just tasks. Ideally would like a list in chronological order showing each task’s start/due date.
Did you try another app like aCalendar+? OpenTasks must still be installed because it provides the storage backend, see illustration on https://www.davdroid.com/faq/system-integration/
OpenTasks shows DUE dates here. I have created a task called “test” with Thunderbird 60.2.1 and Lightning 18.104.22.168, and after synchronizing the task with DAVdroid the DUE date is shown as you can see in the screenshot: 0_1541065122890_a.png
However, please note that DAVdroid doesn’t have any influence on OpenTasks and how it shows tasks. For questions about this, please contact OpenTasks support.
Next, I notice on Android each task will notify upon start time & due time. However on Thunderbird/Lightning there’s no reminder at all. I have to manually edit it & set the Reminder field to 0 minutes  which then enables a ‘start’ time reminder- still not at ‘due’.
Again, this is not managed by DAVdroid, which just synchronizes the tasks between the CalDAV server and OpenTasks. For questions about this, please contact OpenTasks support.
Lastly, whare is the tasks data being stored? I don’t see any of them listed on my FastMail account. Of course the calendar there doesn’t support displaying tasks like Google & Thunderbird/Lightning, so I don’t expect to see them inline on the calendar there like I do on Thunderbird/Lightning. I hope its actually there rather than on someone else’s server.
Does that help?