Should I contact my half sisters

I have a half sister!

Oh how nice!
Assuming she is logged on to Facebook or can be found at all, I would definitely not just send her a friend request.

That would seem kind of strange to me, like you do with a loose party acquaintance. But I'm not a passionate Facebook user, so I can't do too much with that and would be kind of disappointed if my first contact with my sister were via a wordless Facebook friend request.

Does she even speak German? I guess your father's letter was in German, right?

I would do it like taina, if she can be found on FB, namely first a nice, short message, without request.

If she is not at FB, I would write her a nice, short letter in which you simply say that you were happy to learn from your father that you have a sister and that it would be very nice if you would get to know you.

Then you can tell a little about yourself. I wouldn't overwhelm her with questions, maybe she's just a little overwhelmed with the information about the existence of her two siblings.
Then you can only wait.

Even if a lot of things seem to have gone wrong with you in the past, I think it's very nice that your father has now written this letter.