Have you been able to load the font files in page at all in a web browser?
This may be to do with the server that is hosting your font files not being CORS (Cross-Origin Resource Sharing) enabled. This setting is configured server-side and web browsers require this to be enabled for loading font files (among other resources) when requested across domains. Thus for web fonts to load in email, this setting must be enabled.
This is a measure taken by web browsers to support font licensing. The setting can be configured in a HTACCESS file, among other methods. However be sure that the licensing of the font permits this usage.
It may have to do with with your local system settings preventing images served via HTTPS from loading in the Outlook application.
I've encountered this before and the following resolved the issue. Worth a try:
If Outlook 2007, 2010, or 2013 displays some - but not all - images, you may need to change the setting in your Internet Explorer web browser or desktop software.
To change the setting in Internet Explorer:
Open Internet Explorer.
Go to Tools > Internet Options.
Click the Advanced tab.
Scroll down to the Security settings list, then uncheck the "Do not save encrypted pages to disk" option.
Before the changes take effect, it may sometimes be necessary to clear the cache in Internet Explorer and close Outlook (File > Exit to close fully) and relaunch Outlook.
After testing this further on an iOS device running the latest available version of the application it appears that the rendering screenshot was generated from a previous/earlier version (with presumably differing HTML/CSS support).