Nowadays camera batteries are (almost) invariably LiIon (Lithium Ion). There may be a very few specialised exceptions.
LiIon capacity falls as you approach 0 C - it's not that they discharge quickly as such, but that the capacity falls.
LiIon is unlikely to suffer permanent damage under charge or discharge down to 5 degrees C.
Discharge is safe down to -10C or -20C (manufacturers may specify one or other limit) but available capacity is getting extremely low by then.
However, LiIon batteries must not be charged in a normal manner at below 0 C and it is safer to only charge at 5C and above. Charging at normal rates below 0C can result in metallic Lithium being plated onto the anode. This is not removable by subsequent charge/discharge cycles and increases the risk of catastrophic battery failure due to mechanical stress or shock, and decreases battery capacity.
Batteries can be safely [tm] charged below 0 C but only at very low rates. For example, charging at -30C may be safe at rates 1/50th of normal maximum.
Most of the above information comes from or is given in the 'Battery University' article BU-410: Charging at High and Low Temperatures. Additional useful comment is provided there.
Also see - Wikipedia Lithium-Ion Battery - extreme temperatures