Working with dates is usually a headache, and for that, we have tried to keep it as simple as possible. That's the way Exporter is managing dates:
Timezone
Dates will be exported with the timezone of the user who's doing the export process.
We're working with this as Jira is doing, you don't have to configure anything else.
Format
On Jira Cloud you're not able, at the moment, to configure the look and field of date fields. So, for that, we're using the default Jira renderers for date fields.
According to this, dates will be exported with "dd/MMM/yy" format and datetimes with "dd/MMM/yy h:mm a" format.