There's two questions here, sort of. One is about scheduling, and the other about follow-ups.
On the scheduling, it's pretty much a "no". It would seem only a small fraction of Atlassian's users want this, so although they've not closed the requests to do it, they don't seem to be high on the list. There are also improvements creeping in that help with the workarounds - for example, if you write a service to do something regularly, you used to have to say "run every x seconds", but now you've got a more 'cron' like timing for it. And, of course, someone wrote a plugin: https://marketplace.atlassian.com/plugins/pl.com.tt.jira.plugin.theschedulerpro (there may be others, I just picked on the one I've tried)
For follow-ups, there's two simple answers to start with - using subscriptions is a "remind people stuff need doing" and then "escalation" is a bit more firm, as it can change the state of issues, trigger emails, make comments etc - https://confluence.atlassian.com/display/JIRA/Jelly+Escalation . And, again, it's worth a rummage through the marketplace for more.
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