The Crag Cat can turn chain lightning
Why it works
The turning ability contains the text:
The cat has advantage on saving throws against any spell that targets only the cat (not an area). If the cat's saving throw succeeds and the spell is of 7th level or lower, the spell has no effect on the cat and instead targets the caster.
I take the "(not an area)" to mean that chain lightning can work with this ability as it is not an area of effect spell but a targeted one.
If the cat is the target of a residual bolt - can the residual bolt reflect back toward the caster?
When targeted by a residual bolt the cat is the only target for the spell. This qualifies for spell turning and should reflect back toward the caster.
If the cat succeeds the initial save - does the spell reflect to the caster and then target several others, which could potentially be the cat again?
The initial spell would reflect back toward the caster, there is nothing to say the caster isn't a valid target for the spell. Therefore is could leap onto to other targets. Some DMs may rule you need to pass a concentration check or the spell fails after you take damage and before it leaps to another target.
Whether or not you can target the cat again would vary by DM. I would probably rule that you can't, as "A target can be a creature or an object and can be targeted by only one of the bolts".
Who decides where the spell jumps to should the cat successfully reflect the initial bolt - the cat or the caster?
The caster decides who the spell leaps too. The cat is only reflecting the spell, not controlling it.
Does the cat get another reflecting save should a secondary bolt target it after the initial reflected bolt?
The cat would get a save each time it was targeted by the spell. It's Spell Turning feature also doesn't require an action and can be used on every save so it could also reflect additional bolts. This is another reason why I would rule that you could not target the creature again with the same casting.
Range and targeting
It is important to note that the Crag Cat's ability says "targets the caster" not "caster takes the damage" this means that the spell has no effect if the caster is not a valid target.
For chain lightning it has an initial range of 150ft. If the Crag Cat is targeted directly they can turn the spell back as the caster must be within this range to target them.
For each additional bolt however the range is 30ft from the initial target. If the caster is not within 30ft of the initial then the Crag Cat's ability will cause the spell to have no affect on it, but not target the caster as they are not a valid target.