Is the existence of normal numbers evidence true randomness exists, and in every possible world at that? Another mind blowing fact is that most of real numbers are normal, so the normal numbers is not a minority at all.
"True randomness" and "exists" are two concepts that are notoriously hard to pin down, so it is hard to give a straight up answer, but here are two directions of thought.
You can consider the idea that the digits of normal numbers conform to the formal results of probability theory; that's what it means to have proven that they exist mathematically. But now, what you've done is established that this one type of formal/conceptual objects stands in a particular relation to these other formal/conceptual objects. Is that existing? If you are a mathematical realist, then sure, they exist. But if you are a mathematical realist, you probably (ha!) don't have a problem with the existence of (formally defined) randomness in the first place. If you reject the "existence" of purely formal objects, then this formal relationship between concepts won't impress you much.
Sticking to the real world. Given one of these numbers we could construct a seeming perfect random number generator -- a machine/program that can construct more and more of the digits of one of the numbers and provide it to you. But is this really random? Some people could argue that it's not really random since the particular sequence of digits constructed by this machine was fixed (determined) by the normal number that was used to "seed" it. Conversely, if you take a more empirical approach, you might contend that its output is random, since by any computable measure it "looks random".
No. Numerical distributions and randomness are not related concepts, we just often use them together.
If I have a group of one-hundred-thousand people and they are measured and their heights are normally distributed, there is no randomness involved, there is only counting and measurement. Their heights may be deterministic by nature or not, but at the time of measurement, they surely aren't random, they have the values that we have found them to have. The distribution has nothing to do with chance or error.
The same goes for infinite sets -- however their digits are arranged or distributed, they simply are so. There is no randomness involved, only computational assumptions.