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There are two things which may be causing this.

Permissions all by themselves mean pretty much nothing. The combination of ownership and permissions does mean everything. That said, 755/644 permissions are adequate for the web server to serve the files. If you had any permissions issues, that would manifest itself as an inability to install extensions. The solution would be rather simple: enable Joomla!'s FTP Options in the Global Configuration. After installing an extension, run the Fix Permissions again, especially if you have a Plesk-based server. On Plesk-based servers, the default permissions for uploaded files and directories are 0600 which, of course, do not allow the web server to serve the files to the visitors of your site. In all fairness, this is the exact reason why that feature went into Admin Tools in the first place!

The other thing you have to bear in mind is that the default .htaccess Maker settings are purposely set too tight. One of the things you might want to disable is the "Anti-leech protection for static resources outside images/stories" as it may interfere with some extensions (especially if your site can be accessed as www.yoursite.com and yoursite.com - with and without the www).

The other option which may interfere with extensions is the "Front-end protection". The rationale behind this setting is that, since you are using Joomla!, no rogue PHP file should ever (EVER!) be accessed, except Joomla!'s index.php. Why? Because even if a hacker would upload a malicious PHP script due to a component's unpatched vulnerability he'd have no way to access it. In an ideal world where all developers would be security conscious or, at the very least, competent that would indeed be the case. In the Joomla! world where the narrow profit margin forces the use of cheap and inexperienced developers, this is not the case. I've seen many extensions (free and commercial) use their own rogue PHP files, scattered throughout the Joomla! folders. Even worse, I've seen developers serve raw XML files over the web browser (that's a big security no-no). In order to let such extensions work, you have to follow the documentation's instructions for identifying and applying extensions. Either that, or turn off the Front-end Protection, but that is like flying USS Enterprise through a meteor field with shields down: you'll be sitting on a timebomb.

As I've written in the disclaimer (in a much more blunt form), security is not easy and it takes more than a component to achieve. "Fire and forget" security extensions are the fabrication of a vivid imagination of some marketing guy. That's why we never claimed that Admin Tools "is the only protection you need" like some other guys. In the real world where we live in, web application security is a process which you have to continuously adjust and practice. More so whenever you modify something substantial in the application, like installing or upgrading a Joomla! extension or Joomla!'s core itself.

To cut a long story short, here's what you have to do:

1. Go to Joomla!'s Configuration page and enable the FTP options. Make sure you supply the FTP username and password, as well as define the correct FTP path to your site's root.
2. After installing ANY extension, click on Admin Tools' Fix Permissions button.

The problem you are experiencing has NOTHING to do with Admin Tools. It's the way your server is set up. Admin Tools can help you work around the limitations of your server setup with its Fix Permissions feature.

Don't want to believe me? Try something for yourself. Uninstall Admin Tools completely. Uninstall eXtplorer too. Then reinstall eXtplorer. You'll see that the icons, once more, do not display due to wrong permissions. Since Admin Tools doesn't run, which piece of your site software stack do you think is responsible? The web server. QED.