If it has three equal roots, it has only one x-intercept. The function $f(x) = \sqrt[3]{x}$ has only one root, at $x = 0$.
The statement is true, by Rolle's Theorem (which is really a more specific version of the Mean Value Theorem). Say $a, b, c$ are roots of the function, then $f(a) = f(b)$ and $f(b) = f(c)$. Threfore, there are points $i \in (a, b)$ and $j \in (b, c)$ where f'(x) = 0, i.e., the function's slope is $0$.
As for your second question, I think yes, but I'm not sure. Wikipedia's definition of local maxima seems to include global maxima as a special case (you can think of it as taking $\epsilon = \infty$), so I'm going to go with yes.