Yes, the ring acts as CL 3. The save DC for the animal's Will save would be 10 + 1 (the spell level) + 1 (the minimum casting stat modifier needed to cast Charm Animal) = 12.
The wizard ONLY uses his own caster level when activating a staff. (Or a wand, if he has the wizard discovery "staff-like wand").
Otherwise, use the item's CL.
The CL of a magic item is also it's caster level for any spells it casts unless it states otherwise. Note: I do not remember seeing an example where it states otherwise but I guess it could happen.
Staffs are an exception to this in that they use the listed CL or the caster level of the user whichever is greater.
There are some class abilities that let you ignore that. Like the Songhealer archetype for bards, who can use Cure X Wounds wands at their own caster level a few times a day.