Yes, that's true of automatics as well. If you suspect the brake shoe or cabling could get wet and freeze, you should avoid leaving the handbrake engaged.
Yes, at least in the manual case I described above. The issue is that the braking mechanism itself will get stuck if it's left engaged and then gets wet.
Classic handbrakes use a braking wire and don't use the hydraulic breaking system (so it's somewhat redundant, actually). Also, the brake fluid is inside the system and can't prevent the brake cylinder or the brake pads from seizing.