I also have been told that once you have one (and you're on file as having one) that allows searches to your house without warrants.
Absolutely correct. You voluntarily surrender your 4th Amendment rights, just so the feds can come and insure that you have current possession of said firearm or device (suppressor) and they will check the serial numbers!
Of course the plain sight rule still applies and if they see anything that suggests you may be into something unlawful....it will be used against you in a court. That's one important reason I would never go the classIII route. Not because I'm some kind of BG, but because I would dislike having feds come knocking at 0300HRS.