To prove that a mathematical statement is false is it enough to find a counterexample?

When considering a statement that claims that something is always true or true for all values of whatever its "objects" or "inputs" are: yes, to show that it's false, providing a counterexample is sufficient, because such a counterexample would demonstrate that the statement it not true for all possible values. On the other hand, to show that such a statement is true, an example wouldn't be sufficient, but it has to be proven in some general way (unless there's a finite and small enough number of possibilities so that we can actually check all of them one after another).

So logically speaking, for these two specific examples, you're right — each one can be demonstrated to be false with an appropriate counterexample. And both your counterexamples do work, but make sure that the math supporting your claim is right: in the first example you computed $|a+b|$ incorrectly.

By the way, the reference to the triangle inequality is a good touch, but it doesn't prove anything. Rather, it's a very strong hint that suggests that there have got to be examples when the inequality rather than equality holds.

If a statement ${\cal S}$ is of the form "all $x\in A$ have the property $P$" then a single $x_0\in A$ not having the property $P$ proves that the statement ${\cal S}$ is wrong.

But not all statements are of this form. For example the statement ${\cal S}\!:\>$"$\pi$ is rational" cannot be disproved by some "easy" counterexample, but only by means of hard work.

Yes, any counterexample will do. It's often instructive to look for the simplest counterexample. For example, take the one-dimensional vector space $\Bbb R$ and the vectors $a=1$ and $b=-1$ in the case of the first statement.