Yes.
Do mentally incapacitated individuals need a savior?
Yes again.
Neither are capable of sinning. Are they left out in the cold?
No, they are not left in the cold, and yes, they are capable of sinning. All humans are born with a sin nature, born sinful. If you hold any other belief, then you start to come up with a sort of "age of responsiblity" sort of problem; "Well, children don't know any better so they aren't sinful until they're, say, five." Unfortunately, this is both Scripturally and logically unsupported.
Besides, if we say that you have to be responsible before you can be accounted sinfully, you start getting really nasty ethical questions that don't pop up if you simply hold that everyone, from the oldest man or woman down to the newly-concieved child, needs a Savior.
"I have myself seen a small baby jealous; it was too young to speak,
but it was livid with anger as it watched another infant at the breast"
(Book I, Chapter VII, trans. F. J. Sheed).
As I understand it, the Catholic teaching is that all (except Christ and his mother) are conceived with "original sin", not born with it.
If you hold any other belief, then you start to come up with a sort of "age of responsiblity" sort of problem; "Well, children don't know any better so they aren't sinful until they're, say, five." Unfortunately, this is both Scripturally and logically unsupported.
There is no problem with "age of responsibility." Basically, at a certain point in development, a human can choose between good and evil. It they choose evil they have sinned. WHEN that age is reached, only God and that particular person know.
I find no problem with the logic. As far as being "Scripturally supported", I am sure that if a person wanted to, they could assemble a string verses to pretty much support or shoot down any idea.
St. Augustine in his Confessions rejects the notion of the innocence of babies:
"I have myself seen a small baby jealous; it was too young to speak, but it was livid with anger as it watched another infant at the breast" (Book I, Chapter VII, trans. F. J. Sheed).
Augustine had some radical ideas concerning infants. The Catholic Church doesn't accept all of his writings.