What is the easiest way to get a 4.0 version of CiviCRM up to date or at least to a secure version? Is it possible to import a MySQL dump from 4.0 version into a new clean 4.7 version?
At minimum you would need to upgrade to 4.4 - this is the current LTS version and is the oldest version which still receives security patches.
As Parvez recommended, upgrading in stages is sometimes smoother, although 4 versions is not such a big jump that you can't necessarily do it all at once. Always back up your database first, and if jumping straight from 4.0 -> 4.4 doesn't work perfectly, try 4.0 -> 4.2 -> 4.4.