Hi Egbert,
As for your first question, yes you can export the list to mailchimp. What you can do is that, go to your database, export the #__easyblog_site_subscription table or any table into csv file. Then go to your mailchimp, and import this and sync the user accordingly.
You can use this as a guide
http://kb.mailchimp.com/lists/growth/import-subscribers-to-a-list
For your second question, yes. what EasyBlog does with the mailchimp is that, when somone subscribe to your blog, EasyBlog add the subscriber to your mailing list defined in mailchimp integration.
Have you setup a cron job for your site? We highly recommend that you setup the cron job so that the email notification can be send smoothly. If not yet you need to enable send email on page load. You can find this settings at your backend, under settings -> notifications -> general -> email processing