Hmmmm...I never saw a furnace filter that didn't pass some particles. They will filter fibrous materials such as lint, but not fine particles such as soot. Normally, that would not matter, but this time we are talking about radioactivity passing through. I suppose inhaling even a single "hot" particle could initiate a cancerous cell mutation in the lungs. Then there is a matter of the fan. It takes a strong motor to pull a volume of air through a fine filter. We're not talking batteries here, but A/C current. After a nuclear blast, what are the odds the power plant, lines, substations, etc. will still be intact?
Depends on where you are, how wide spread the engagement is, etc, etc. The chances of the power grid being up and you being in the fallout zone from an Al Queda dirty bomb is a lot greater than they would be in say an all out nuclear exchange. As far as the filters go, think Hepa filters, which do screen out all but the smallest of particulates.
I'm just popping off tidbits of learned and gleaned knowledge; some of it decades old. No way you could get the knowledge to survive from a simple paragraph on a conservative blog. You are asking the right questions, though, and sound genuinely interested. There are some really good "survival" titles out there, which go in depth on the issue, as well as other survival situations. The local librarians think I am a bit of a kook, becasue of some of the books I get :-)Read up; having the knowledge and not having to use it it a heck of a lot better than not having it and needing it.
Big question, are you an ant, or a grasshopper? Have a great weekend.