The three red-eyed mutations are lutino, fallow, and recessive silver (called pale fallow in Europe). Lutino is common and the other two are rare. A whiteface lutino is an all-white bird with red eyes, and it looks like the bird in the foreground of your first picture is one of these.
But a bird with a lot of grey on it is NOT a lutino. If the other birds in this picture have red eyes then they are probably recessive silver - they don't look like fallow.
A bird that is visual for any of these mutations will have red eyes, and a bird that is not visual for any of these mutations will have dark eyes. So the outcome will depend on what you breed your red-eyed bird with. For example, if you breed a lutino female to a pied male who does not have the lutino gene at all, the babies will all have dark eyes because you will not get any lutino babies. If you breed that same lutino female to a pied male who is split to lutino, you will get some lutino babies and they will have red eyes. The non-lutino babies will have dark eyes.
A red-eyed bird is not necessarily weaker than any other bird. If you get a lutino that was produced by a bird mill it is likely to be weak, since they use bad breeding practices to make as much money as possible. If you get a lutino from a breeder who follows good practices, it should be as strong and healthy as any other cockatiel. The same thing applies to all the other mutations too. Bad breeders tend to produce weak birds, and good breeders tend to produce strong birds.