How to extract specific emails from Exchange Online using PowerShell and move or sort them?

You can't use Export-Mailbox or New-MailboxExportRequest cmdlets on Exchange Online,

The only way is Exporting it from Outlook client and use a Date Filter to extract the specific date range you want.

Update:

You can use the Compliance Center in the Office 365 Admin Center to search and export content with filter options etc.

In addition to @avshalom there are a couple more solution I can think of.

If you have E3 (or Exchange Plan 2) you can leverage eDiscovery. You can create rules to meet the above requirement and copy these to a eDiscovery mailbox and/or export to PST.

Now - if you want to get creative, you may be able to leverage something like IFTT or Zappier to handle interaction over O365 API's. I've had not looked at how granular these rules can be set (per date range, etc).