No to the first question and the second question
With respect to the first question, Bigby's hand (PHB p218) contains the following text.
You create a Large hand of shimmering, translucent force...
Compare this with the text of firebolt (PHB p242)
You hurl a mote of fire at a creature or object within range.
Or Toll the Dead (PHB p169)
You point at one creature you can see within range, and the sound of a dolorous bell fills the air around it for a moment. The target must...
Or Disintegrate (PHB p233)
A thin green ray springs from your pointing finger to a target that you can see within range. The target can be a creature, an object, or a creation of magical force...
All of these spells explicitly mention targeting only one target and should be valid spells for use with War Caster. Bigby's Hand creates a magical effect on the battlefield that may be used in a way that targets only a single target, but it does not pertain to War Caster as it is not a spell that
target[s] only that creature.
With respect to the second question, War Caster states the following (PHB p170)
...you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action...
As you are not using a bonusan action and/or casting a spell, using an effect from Bigby's hand that has already been cast does not qualify for use with War Caster.