Try to double or triple or even x10 the number of dots on your piece of paper and you get "patterns" forming. Repeat the experiment 1 million times and you might find other patterns appearing.
I love thinking about the concept of randomness, too. Not so much from a mathematical perspective though.
When you flip a coin, the air, the way you flip, the way the coin is manufactured, all affect the outcome in ways we can't really calculate. When you roll a 10-sided die or 6-sided one, the manufacturing of it, the way some sides might have more density than others. The way a "10" is grooved means that side is ever so lighter than the side that has a "1". For sure all these affect the outcome, and, therefore, have a say in how random the coinflip or dice throw is. I've seen people manipulate dice throws, coin flips, measuring exactly how much strength to flip the coin, ensuring every throw has the same number of flips.
The way a dealer shuffles. The way a casino card stack is cut, the way the roulette wheels are oiled. The way the metal ball hits when it's thrown on the wheel.
How random is random? It's a lovely question!
I like the way you explain things that affects its random outcomes, but what if we consider randomness of a computer system? All physical tangible things could be impacted by their physical characteristics on how they were created. What about the computer system who uses random generating functions, do they have any basis at all? Are they really random? How come they are programmed to provide random numbers if computers are precise and absolute.