They lose some weight but that is natural.
They don't lose a lot of weight because they are not very active but they do lose some.

Yes - that is brumating. It is necessary for many species to succesfully breed, I'm not sure if it is necessary for alligator lizards to succesfully breed or not.

The room I keep them in over winter cools down to mid 50s. I don't know how cool they need to get to succesfully breed.
-----
3.6 L. getula californiae - 19 eggs (Cal. King)
1.1 L. getula nigrita (MBK)
1.0 Pantherophis guttatus guttatus (Corn)
0.1 Pituophis catenifer catenifer (Pacific gopher)
3.3 Elgaria multicarinata multicarinata (Cal. Alligator Lizard)