I want to know if the Leibniz differential notation actually leads to contradictions - I am starting to think it does not.
And just to eliminate the most commonly showcased 'difficulty':
For the level curve $f(x,y)=0$ in the plane we have $$\frac{dy}{dx}=-\frac{\dfrac{\partial f}{\partial x}}{\dfrac{\partial f}{\partial y}}$$ If we were to "cancel" the differentials we would incorrectly derive $\frac{dy}{dx}=-\frac{dy}{dx}$. Why does this not work? Simple: The "$\partial f$" in the numerator is a response to the change in $x$, whereas the "$\partial f$" in the denominator is a response to the change in $y$. They are different numbers, and so cannot be cancelled.
Related: consult the answer to this previous question.
The other part has been moved to a new post here.
Answer
As you suggest in your own question, there is in fact no contradiction in Leibniz's notation, contrary to persistent popular belief. Of course, one needs to distinguish carefully between partial derivatives and derivatives in the notation, as you did. On an even more basic level, the famous "inconsistency" of working your way from $y=x^2$ to $dy=2xdx$ is handled successfully by Leibniz who is aware of the fact that he is working with a generalized notion of "equality up to" rather than equality "on the nose". These issues were studied in detail in this recent study.
The formula $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$ holds so long as we assign to the independent variable $du$ in the denominator of $\frac{dy}{du}$ the same value as that given by the dependent variable $du$ in the numerator of $\frac{du}{dx}$. On the other hand, if as is usual one uses constant differentials $du$ in computing $\frac{dy}{du}$ the formula will be incorrect. In each instance one has to be careful about the meaning one assigns to the variables, as elsewhere in mathematics. For details see Keisler.
The OP reformulated his question in subsequent comments as wishing to understand how Leibniz himself viewed his theory and why he believed it works. This seems like a tall task but it so happens that there is a satisfactory answer to it in the literature. Namely, while Leibniz was obviously unfamiliar with the ontological set-theoretic material we take for granted today, he had a rather clear vision of the procedural aspects of his calculus, and moreover clearly articulated them unbeknownst to many historians writing today. The particular paradox of the differential ratio $\frac{dy}{dx}$ being apparently not equal on the nose to what we expect, e.g., $2x$ (which in particular undermines the "tautological" proof of the chain rule in one variable) was explained by Leibniz in terms of his transcendental law of homogeneity. On Leibniz see article1 and article2.
The consistency of Leibniz's law is demonstated in the context of modern set-theoretic assumptions in terms of the standard part principle.
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