In addition to seconding Sebastian's recommendation to upgrade both the DNN framework and the media module, I suspect that the direct cause of the failure to display media is related to a manual change that must be done to the web.config when changing to an ASP.Net 4.0 application pool:
In the site's web.config file, locate the <handlers> section. In it you will find several nodes similar to this one for the LinkClick handler:
<add name="LinkClickHandler" path="LinkClick.aspx" verb="*" type="DotNetNuke.Services.FileSystem.FileServerHandler, DotNetNuke" preCondition="integratedMode,runtimeVersionv2.0" />
On each node in the <handlers> section where you see preCondition="integratedMode,runtimeVersion2.0" change the attribute to
preCondition="integratedMode"
This necessary change did not become an automatic one until either the late DNN 5.6.x versions or 6.x. Without it any use of the LinkClick handler (used by the Media Module, the CaptchaHandler, etc. will result in the missing file error you are seeing.