Apparently the problem you have is not related to Akeeba Backup, but file/directory permissions and your host.
First, let’s tackle the uninstallation. Akeeba Backup is a WordPress plugin. Like all WordPress plugins all you have to do to uninstall it is delete the directory it installs in. In this case it’s wp-content/plugins/akeebabackupcore (the name of the directory could be different as it’s up to you or WordPress to define it, not us).
Then, let’s talk about updates. You can’t update because WordPress can’t write to the directory of the plugin. You probably have the same issue with other plugins. Changing the directory permissions is one way to work around the issue. The other way is downloading the plugin, extract the ZIP file locally and upload by FTP the files into the plugin’s directory, i.e. wp-content/plugins/akeebabackupcore (the name of the directory could be different as it’s up to you or WordPress to define it, not us).
Finally, the backup. As you have already said the problem is with permissions. Akeeba Backup does not change the permissions of the backup directory. Your host or another plugin installed on your site does. Please ask your host. If they don’t change the permissions there’s an easy workaround to your problem which, by the way, it’s the recommended, secure practice in our documentation. Create a new directory outside your we root (above public_html). Then go to the Configuration page of Akeeba Backup, find the Output Directory setting, click on the folder button next to it and browse into the directory you have created above public_html. Then click on Use, Save & Close and you’re ready to rock and roll.
