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There is nothing to stop you creating a sub-task type called "task", although you will need to rename the parent-level "task" to something else first.
By default, JIRA Software has a sub-task type of "technical task" which is a way to avoid the naming clash, but you should feel free to use whatever names that work best for your organisation!
You can not add a task under a story. You can add sub-tasks and that's it.
Not sure if there is an add-on in JIRA cloud or not but JIRA Server has options like Structure which can bring in a hierarchy inside JIRA.