Yep, it is very likely a false pregancy. If she ended her heat in late November, this is around the time she'd have whelped had she conceived. Alas, following a heat cycle a bitch will go through all of the bodily and hormonal changes you'd expect from a pregnacy, whether bred or not. That's a major reason it is recommended that non-breeding dogs be spayed before the first heat. You could do a search here on 'false pregnancy' for more information.
Poor girl - well there's not a lot you can do except let her ride out the hormonal storm, then have her spayed straight after.
As for the mange thing, yes your vet is quite correct. Not only would a pregnancy and raising a litter put excessive strain on her immune system and likely bring the mange back on, but it is indeed hereditary and would be passed to any puppies. Great to hear you've decided to have her spayed

It's really in her best interests, and of course, the interests of the breed as a whole. I wish more people would be as responsible
Here's a site with some good information on Demodicosis (demodectic mange)
http://www.upei.ca/~cidd/intro.htm (scroll down to inherited skin disorders, click that, then scroll down to demodicosis)
PS: Yes, her mother should also be spayed, but so should her father be neutered - unless it is known which one of them is the carrier. Demodicosis is genetically inheritable - so that means from either or both parents
