For the bulk close and bulk move you could write a plugin and use the job scheduler to run it as a cron. There's a guide here. Another way to do would be to define a service in ScriptRunner that would call a groovy script every 2 weeks. There's an example here.
For the automated email, create a filter that searches for all issues that are in an "awaiting input" state (however you define that) whose status hasn't changed in 1 week and the assignee is currentUser(). Share the filter and subscribe all users to that filter with the caveat that if no issues are returned the mail shouldn't be sent. Since the filter will be run separately for each user, only users who have such an issue will get the nag mail.
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