Film degradation is basically chemical reactions.....which are speeded up by a number of factors....or slowed down.
Chiefly temperature....all chemical reactions including what happens to photographic emulsions over time are speeded up by higher temperature. So freezing is of huge benefit. Added to this the freezer is dark, which can only help compared to, say, storing your films on a sunny windowsill for 30 years.
Film speed is also a big factor. The higher the ISO, the faster the damage.
B&W film lasts much longer than colour film.
As for background radiation, the lead container should see to that. The metal casing of a fridge will mitigate this somewhat too....especially any alpha particles.
Cosmic rays are a different matter because they are made up of all manner of high energy particles including electrons, various nuclei some of which are radioactive partoc;es and some are not. Lead will protect against the radioactive particles but not the others. However, the damage is cumulative (ie happens over time) and is very slow because the concentration of cosmic rays is not high. My own work with radioisotopes includes demonstrating the shielding effect of various materials with alpha, beta and gamma sources....a good 5mm of lead will cut out all of the alpha and beta, and 90% of the gamma.
In practise, B&W film can be frozen for 40-50 years and come out fresh as the day it was purchased. Colour negative film at least 20 years. Colour reversal is a bit of a crap shoot but even at 20 years old you'll get images even if the colours have shifted.
There are people out there shooting 100 year old glass plates which haven't been refrigerated and getting decent photos.
Additionally....everything
@reddesert said.
The only other caveat is humidity. Store film, especially roll films with backing paper, in their sealed foil containers from the factory. Do not open them until the film has come up to room temperature, at least an hour after the freezer. I often leave overnight to be sure. Even then, some of the bad batches of film produced in recent years might have backing paper issues.