This is most likely due to the Subscription not being set to use php in fastcgi mode (i.e. it is using mod_php instead).
What happens in mod_php mode is that any files/directories created by a php script are owned by apache. These files can't then be modified/uploaded to by the "ftp username" via Dreamweaver. Similarly, Plesk's File Manager only allows files/directories owned by the ftp username to be edited/uploaded to.
All the files and directories should still be viewable in Dreamweaver (Remote site, not Local). I'm guessing that the "other" attribute on some of those directories is not set to allow reading or something like that, as otherwise you'd at least get a copy of the file locally, even if you can't edit it remotely.
The solution to this is to change to running php via fastcgi, and also to recursively chown the directories and files to be owned by the ftp username (note: do make sure you include any .htaccess files in the chown).
There is some small risk of screwing things up when doing this, so make a backup first.
You'll note that the permissions on some directories and files in the existing site may look a bit odd -- they are 757 or 747 or something. This is to allow the php script (apache) to create files and subdirectories (e.g. the upload directory) in mod_php mode without encountering write/create file permission problems. Most of these odd file permissions are not required when running php in fastcgi mode, but altering them could be a bit tricky as it isn't always obvious what directory needs to be set to what, so it might be best to leave them as they are (though there is a small security risk involved).
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