How can I import e-mails to CiviCRM?
Are you using MS Outlook or Gmail? We've written a couple of extensions/plugins which allow the functionality you've described.

A couple of posts will give you some background to the project

https://vedaconsulting.co.uk/outlookforcivicrm-release
https://vedaconsulting.co.uk/outlookforcivicrm-civicase

We've also recently done some work allowing you to sync CiviCRM Groups to mailing lists within your outlook client - blog post to come.

The CiviCRM extension and the outlook plugin can be found in our git repo along with installation instructions.

https://github.com/veda-consulting/uk.co.vedaconsulting.outlookapi

Thanks Erik and Pete for the nudges!