I use a cron script that pokes things in over the REST API, but for something a bit less reliant on a bit of command line stuff, yes, you can do it.
There are several add-ons in the marketplace that can do it with a friendlier UI from within Jira, and the generalised tools like Automation and ScriptRunner enable you create regular processes.
You can, of course, write your own too. Have a look at https://developer.atlassian.com/docs/atlassian-platform-common-components/shared-access-layer/scheduling-events-via-sal-tutorial if you want to do it yourself.
Don't use SAL's scheduler - it's in the process of being deprecated. There's now a new scheduler at https://bitbucket.org/atlassian/atlassian-scheduler which replaces SAL's scheduler. There's an example of how to use it at https://bitbucket.org/cfuller/atlassian-scheduler-jira-example
Yes it is possible to create recurring tasks in Jira. You can find a tutorial in this page:
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