I believe Wolfram is throwing an error because $\int_{-\infty}^\infty$ can be interpreted in different ways. You cannot treat $-\infty$ and $\infty$ the same way you treat numbers. Usually in calculus (and physics), infinity shows up in the context of taking some limit as some quantity gets large without bound. The trouble is infinity can be approached in different ways and at different rates ($x^2$ approaches $\infty$ much faster than $x$ for example). And these different ways you can approach infinity are not necessarily equivalent. Thus you get conflicts like "What is $\infty - \infty$?" which can have any number of answers because you can approach one infinity faster than the other.
Because we can approach $\infty$ in different "ways" we must pick a way when we define something like $\int_{-\infty}^\infty$. The standard way is to define it like so:
$$ \int_{-\infty}^\infty f(x)\,dx := \lim_{a \to -\infty} \int_a^c f(x)\,dx + \lim_{b \to \infty} \int_c^b f(x)\,dx $$
where $c$ is a constant and can be any real number. You'll notice that if we try to apply this definition for $\sin x$ we run into a problem. For sake of clarity, let me just focus on the first term of the definition (the one with the limit in $a$):
\begin{align}
\lim_{a \to -\infty} \int_a^c \sin x\,dx &= \lim_{a \to -\infty} \Big[-\cos x \Big]_a^c \\
&= \lim_{a \to -\infty} \big[\cos(a) \big] - \cos(c)
\end{align}
But this limit doesn't converge because cosine oscillates forever when you approach infinity. Since this limit is a component of the definition I've stated for $\int_{-\infty}^\infty$, it means the expression $\int_{-\infty}^\infty \sin x\,dx$ must be left undefined, and hence Wolfram|Alpha (correctly) barfs.
But of course there is another way you can define $\int_{-\infty}^\infty$. Namely as the Cauchy Principal Value that @Basti mentions. With this definition we define
$$ \textrm{p.v.} \int_{-\infty}^\infty f(x)\,dx := \lim_{R \to \infty} \int_{-R}^R f(x)\,dx $$
where the "$\textrm{p.v.}$" is prefixed to distinguish it from the standard definition. Now what happens if we try to use it on $\sin x$?
\begin{align}
\textrm{p.v.} \int_{-\infty}^\infty \sin x\,dx &= \lim_{R \to \infty} \int_{-R}^R \sin x\,dx \\
&= \lim_{R \to \infty} (0) \quad \textrm{because $\sin$ is an odd function} \\
&= 0
\end{align}
which is what you were expecting.
So in short: $\int_{-\infty}^\infty$ can either converge or diverge for some function depending on exactly how you define that expression. Not all sensible definitions are equivalent, so you need to make sure you know which one you're using before you evaluate it. However, as a final note, there are those times when the standard definition and the Cauchy Principal Value do agree. In fact, whenever the standard definition works, the Cauchy Principal Value is guaranteed to converge, and in fact will converge to the same result. The converse is not true in general (as I've shown). Thus the standard definition is a "stronger" notion than the principal value notion: it's harder to satisfy, but it tells you more if it is.
