Hi Matthew,
The Task Scheduler API is part of Windows and is available to any application that wishes to implement it. So yes, Toad can still access the API even if there are tasks that were not created by you. That should have no bearing on Toad being able to create and schedule new tasks.
For the old tasks, you can either disable them or remove them. It's your choice. I would probably disable the old ones until new tasks are created. That way, you can compare them in Windows Task Manager to see if there are any differences between them.
Yes, you can run TOAD from the Task Scheduler. In fact, that's precisely what the Automation Designer in Toad does. You first create an automation script. Then you schedule it, which creates a task in Window's Task Manager. If you inspect that task, you'll notice it is configured to launch Toad and execute the automation script you scheduled.
-John