Is it possible to have an RSS feed module work in Medium Trust with the security limitations that the Medium Trust policies impose?
The problem isn't exactly the News module. When ASP.Net was brought out, Microsoft added more security 'features' to try to stem the tide of growing issues with certain calls, including httpwebrequests, which this module, as well as others use. At first it wasn't really an issue because many hosts were operating in Full-Trust mode at the server level. Then, as more folks understood the code access security model, host providers began to lock down their machines/servers.
The issue you are experiencing has long been a issue for some. Others seek out hosts who understand the CAS issue, and have come up with other ways to negotiate security that allows httpwebrequest to function. PowerDNN is such a site host, others exist too, you just have to ask before you commit.
So, the answer to that question is NO in the strict sense, but YES depending on your hosts knowledge and ability to work with a DNN site.
If not, is there a way to get the module strong named and a PublicKeyBlob generated for it so that I can ask my hoster to make an exception for the assembly to operate in Full Trust?
Yes there is a way to do it, but then your host provider would need to load the assembly in GAC, which most would probably not do, since anything loaded in GAC of a machine/server gets Full-Trust, so you are back to opening up security issues. That type of situation seems rather an unlikely solution.
David - Your issue may not even be connect to code access security. If you are operating in a Full-Trust environment at the machine/server level, then you can get the 'download' error from many causes, which are pretty well documented in the top post on the News forum. You cannot compare the action of the browser to render a feed with the inability of the News module to download it. They are 2 distinct types of applications, one is operating on your machine as a windows app, and the other is a web app, which calls for different methods to use. If you post your URL that is giving you the issue, maybe you can get further help. My advice is to review the 'Post Your broken feeds' post, and post your URL to it for further help.