I don't think the extra slash is causing the problem, but I do think that the directories it mentions are not Writable and they need to be writable by your webserver to upload media.
Can you check the permissions of the directories and make sure they are writable by your webserver?
That sounds like it should be fine, as long as you are sure the www group is what your webserver is running as. The line that performs the check is (actually, there are 3 of them...):
envcheck.php (admin/plugins/mediagallery)
line 149, 166 and 184. You can edit this file and remove the beginning / before mediagallery in that line. That might fix it, but that extra slash shouldn't make a difference.
The other thing to check is to see if you can upload media. The configuration checks are just checks, they do not determine whether or not the uploads will work, they just report what it thinks are errors.
Last, you might change the owner of all the directories below mediagallery/mediaobject (chown -R owner:group *) to be the same as the webserver's running process owner.