Derivative of $x^2$

For the first "paradox", you simply found a way to make it look like the number of terms in the sum is constant. In reality, the number of terms is increasing with $x$. Imagine if you wrote $x=1+1+...+1$ ($x$ terms), then differentiated to get that $x'=0$. But you forgot to take the variation in length of the sum into account. This version is perhaps more transparent because you no longer have the illusion that you took $x$ into account already.

For the second paradox, that formula $y=nx \Rightarrow y'=n$ very explicitly depends on the fact that $n$ is constant with respect to $x$. Otherwise the fact is simply wrong.

The answer lies in the subtle difference between

$$f(x) = x^2$$

and

$$f(x) = x + x + x + \cdots + x (x\ \mathrm{times}).$$

What is this? Well, the first expression is used to define a function on real numbers. The second expression refers to a function on natural numbers. Namely, "x times" only makes sense for $x$ a natural number. What does it mean to add $x$ to itself $\frac{1}{2}$ of a time? It doesn't make sense.

The idea of multiplication as a "repeated addition" doesn't work when you get to multiplying non-natural numbers. It's better to think of a real product $ab$ not as "$a$ added to itself $b$ times", but rather like "$a$ scaled by the scale factor $b$". A scaling operation can be varied in intensity continuously; a process of repetition cannot.

So, not being a function of real numbers, you cannot take its derivative.