The RSS readers work by fetching the RSS XML directly from the third party servers. If the servers do not have a functioning crossdomain,xml file, they'll typicaly be fetched through some sort of proxy, such as Yahoo Pipes that can do the feth on its behalf.
There is no significant writable storage on the classic chumby, so nothing can be stored there.
Widgets run in a security sandbox much like Flash movies running in a browser, and therefore have no access to the local file system, therefore cannot copy or write data to the chumby.
SharedObject (in FlashLite "MobileSharedObjects") are supported, but are stored in RAM, so they're lost on a power cycle.
Widgets that want to save persistent values across executions should store those settings on the server.
As a hack, one can create a database application (sqlite is pretty easy to port) and run in on a USB dongle. The typical way to make this available to Flash is to create a set of CGIs using the built-in web server and use XML or LoadVars Objects to access it.