It appears folders can't be dragged into other folders on any vault. However, you can create a new folder first, then select multiple sites and drag them in.
If you're offline, you should be able to login to the browser extension and access your sites, as long as you've done so at least once in the past. As for making changes, however, this isn't recommended, since the changes need to be synchronized with our servers. If you make changes while offline, we do add them to a retry file and try to contact the server again later, but I wouldn't recommend relying on that, since your retry file could be blown away, etc.