Yes, I think it’s true that all babies have blue eyes, and they get more pigmented as the baby gets older. Though I’m not sure about dark skinned babies, maybe they are born with dark eyes. Sorry, that’s not a very definitive answer!
Melanin, the pigment which gives eyes their colour, hasn’t been fully deposited in the iris (the coloured part of the eye) straight after birth. As time goes on, people who will eventually have dark irises will have more melanin deposited in the iris – whereas people who will have lighter eyes, will have less. Albino people will have no melanin in their irises, and their eyes will look red because of light reflecting off the blood vessels within the eye.
No, it isn’t true that they are all born with blue eyes. It is true that all babies start out with blue eyes though, and the pigement that makes their eye colour comes in later, but that could be before they are born.