As far as I understand it the laws are well defined but they predict a certain amount of randomness in any experimental results. Heisenberg's Uncertainty Principle is probably the most famous example.
Yes exactly.
It is important to understand that the uncertainty is a property of the particle itself, it is NOT randomness because of the measurement (through an error of measurement for example). This means no matter how exactly you theoretically could measure, it is not possible to determine two properties like velocity or position of a particle simultaneously with a high accuracy. As a fact, at small scale, nature becomes more and more indeterminable.
Einstein doubted the results and thought that the theory is actually incomplete and that there are hidden variables, but it was (along with other scientists) also he himself who proved that there are no hidden variables and that the indeterminism at small scale is part of a natural law, a universal property of all particles. His doubts once led to the quote 'god doesn't play dice' which is often quote mined and misinterpreted by religious nuts.