Can I move emails to an external hard drive with the attachments?

You can just drag them from the Mail window into some folder in the Finder as “eml” files. They will retain their attachments and you can double-click them to open them in Mail. This is just an export. You would still need to delete the message in Mail.

These messages will probably not be searchable inside Mail anymore, but you could probably search them via Spotlight.

Try it with a few messages first to see how it works.