No, that is not true. Not at all. Some breeds of dogs have light blue eyes in their genetics, but are not born blind. The eyes can be any color, they're not a certain color every time. Are deaf animals born without ears? No. Mute animals without a mouth? No. It's just a stupid rumor that someone made up because they either really didn't know or wanted to fool people. It's not true, so don't believe it.
Not necessarily, it varies for every animal. Humans, for an obvious example are NOT all born with blue eyes. Many types of domestic animals such as rabbits and dogs, and most cats, however are born with blue eyes.