There are some aviation GPS units that can overlay radar weather data onto the screen of the unit. They can also show other aircraft traffic. The Garmins (GNS 430's) in my plane has the capability but you have to pay a monthly subscription fee to get it. Not sure how the information is transmitted to the aircraft in flight but it certainly isn't wireless internet.
There are many different type of weather "pictures". The radar that shows the thunderstorms is ground based. The photos that show clouds and hurricanes are actual photos taken by a satellite (either geosynchronous or polar orbitting) and transmitted to the earth via radio signal. The only "post-processing" they need is converting from a digital radio signal (analog back in my days as a shipboard meteorologist) to a photo - much the same as emailing a jpeg file. Many of the pictures they show on the news are a composite with the radar image superimposed over a satellite picture.

So, to answer the original question, the capability is there to get more data on the GPS screen but probably not on the low-end junk (my opinion) in the FEH.

The other part of your question regarding fees - there is no fee for the use of the satellites but any good GPS system should at least offer a subscription service to allow for updates of the database. It can get quickly obsolete without it.