<p>Does anyone know if you can and how to start off a scheduled Windows task on a Remote Server from within a SQL Server Integration Services (SSIS) package?</p>

It should be possible as the Task Scheduler has a scriptable
COM API
that can be used for interacting with tasks.
You could therefore either create a custom task that uses COM interop to call the Task Scheduler API, or it'd probably be quicker to use an Active X Script task to do your dirty work.