Everyone in a shared list can see every task in it, whoever the task is assigned to. Assigning a task shows who is responsible. It doesn't hide the task from others.
No. There are no private tasks inside a shared list, and you can't share just one task.
Folders are personal too. Other people never see your folders.
Everyone in the new list can see it, including its comments and files. People who aren't in the new list can't see it any more. See How to move a task to another list.
Click the search box and choose Assigned to me. It shows your tasks from all your lists.
Inside a list, you can also use Sort By, then Assigned, to group tasks by person. See How to sort the Tasks in a List.
Everyone in the list can see all tasks and comment. Only the task's creator, the list owner or a list admin can assign a task, set the due date or add a verifier. See How to manage Sharing in a List.