I would like to know more about Sol's twin(?) named Nemesis and planet X, which is supposed to orbit between Nemesis and Sol. Is there any proof that they exist? Where are they supposed to be located? What is the length of orbit of planet X supposed to be?

There are a few astronomers who think this 2nd sun is around, possibly with another planet, but they are in the minority. One fact on their side is that most solar systems seem to be binary. The period is supposed to be tens of thousands of years, maybe more. I read an articl on it not too long ago. I'll see if I can find it...

I couldn't find the article but you made me curious. I found 2 sites that had interesting info on them: Hypothetical Planets
SPACE.com -- Nemesis: Does the Sun Have a 'Companion'?

Seems I was a bit of on the period. It's more like 30 million years. If Nemesis exists.

Here is a credible website that discusses "Planet X:"

Phil Plait's Bad Astronomy: Misconceptions: Planet X