Suppose $y(x)$ is continuous and $y'(x)=0$ has uncountable many solutions but $y(x)$ is not constant on any interval. Is this possible? Suppose $y(x)$ is continuous and $y'(x)=0$ has uncountable many solutions but $y(x)$ is not constant on any interval. Is this possible?
Take a Cantor set $C\subseteq [0,1]$ and let $d(x)$ be the distance function from $x$ to $C$.  Then $(d(x))^2$ is differentiable with derivative vanishing at all points of $C$ hence uncountably many.
Yes, it is true. A classic example of such an odd function is the Cantor function. It is defined like this: for $x$ in $[0,1]$

1.Express x in base 3.

2.If x contains a 1, replace every digit after the first 1 by 0.

3.Replace all 2s with 1s.

4.Interpret the result as a binary number. The result is c(x).

This function has derivative $0$ almost everywhere (i.e. the places where it doesn't have $0$ derivative has measure $0$) and yet somehow manages to increase in value from $0$ at $x=0$ to $1$ at $x=1$.

There is plenty written about the Cantor function. Perhaps wikipedia would be the best starting place: https://en.wikipedia.org/wiki/Cantor_function