Answer to Part One
Without me you'd be ice, but I'm at the poles:
Without sunlight the Earth would just be a dead frozen ball of ice. Even though the poles are made of ice, the sun also shines here.
You'd meet your demise, but I'm not within you.
We need light to provide heat and energy to the Earth, so without light there can be no life. Light is essential to humans, but it is not an internal part of us.
I'm mixed without mixing and caught without catching,
White light is a "mix" of all the colors of the spectrum, although no "mixing" ever found place. An object can "catch" the light, although it doesn't actually catch it like you would catch a ball.
I'm here when they're hatching.
When eggs are hatching in an incubator, they are often put below a lamp emitting infra-red light.
Answer to Part Two
My sister hides, out of sight,
Radio waves are related to light waves. They are two kinds of electromagnetic energy. Radio waves are not visible to the human eye.
She can't be touched, but felt she might.
It is not possible to reach out and touch radio waves, but some people are said to be hypersensitive to electromagnetic fields from cell phones, radios etc. (Furthermore you can feel the emotional effect of the data transmitted by the radio waves.)
She'll go through you, and you won't know she's there,
Radio waves can pass through our bodies without us noticing.
Unless you are, indeed quite rare.
As mentioned above, a few people are said to suffer from electro-hypersensitivity, but it is a rare condition.