Hello;
maybe your host has severe restrictions on "PHPSUEXEC", so you cannot make desired permissions trough your 4images interface, as the rules say it is forbidden for "apache" user.
One common way is to go to your admin control panel, provided by your host, or simply by FTP client, and each time 4images creates a category, you will have to delete/re-create or simply create it (common issue for Plesk users), without forgetting to create the same category_id in thumbnails, giving it permissions with CHMOD 777, if host allows.
Example:
Gallery category number one:
Delete/Create or simply create folder /your_site/data/media/1/ with CHMOD 777, to be sure you can upload images to category having cat_id=1.
Where 1 is the ID of your category (visible in 4images Administrator Control Panel)
Delete/Create or simply create folder /your_site/data/thumbnails/1/ with HMOD 777, otherwise you will never be able to create any thumbnail.
Keep on goin'