First-level subtasks are moved through the columns on their swimlane. But second-level subtasks, e.g. a task's subtasks, have to be always connected to their parent (e.g. task) and placed on the single card.
As for resolving parent task on when all subtasks are resolved, please use customizable workflow (as example look at the JT-18908).
Is it second-level subtask?
Please attach screenshots.
Also please answer on the questions in my previous comment. Thank you!
As for resolving parent task on when all subtasks are resolved, please use customizable workflow (as example look at the JT-18908).