I think this is the correct way to phrase it. Just because the probabilities of each are both 50% doesn't mean it's more likely than not to get the same number of ones and zeros. It would just mean you're equally likely to get a few more ones as you are to get a few more zeros. But the counts are unlikely to be very far apart.

In the limit, yes.

But for the first few bits maybe "almost".

With n random bits, you will likely have approximately \sqrt{n} more ones than zeros (or vice versa).

You aren't suggesting that something random should be predictable, right?

Yes, almost. They will approach equal as the count approaches infinity.

So if ypu ever see a random pick equate to 50/50, the end of the universe is upon us.

If a random streams length is odd, they'll never be the exact same.