Is $\pi$ equal to $180^\circ$?
Not $\pi$ but $\pi$ radians equal $180°$
It would be reasonable to define that: $$x^\circ = \frac{2\pi}{360}\cdot x$$ in which case yes, $180^\circ$ literally equals $\pi$. Leox's answer is probably a little more correct though.
Addendum. After a bit of thought, I've changed my mind slightly; I no longer think that Leox's answer is correct anymore. To summarize my current beliefs about the issue: $\pi$ literally equals $180^\circ$, both are unitless (as others have argued), and neither degrees nor radians are really units at all (again, as others have argued.) In particular, I think that "radians" and "degrees" are basically systems of conventions, not units like meters or seconds.
Lets discuss this a little. In my opinion, what's really going on is that there is a function
$$\mathrm{AngleInRadians} : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow [0,\pi]$$
given by
$$\mathrm{AngleInRadians}(v,w) = \mathrm{arccos}\left(\frac{v \cdot w}{\|v\| \cdot \|w\|}\right)$$
and another function,
$$\mathrm{AngleInDegrees} : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow [0,180]$$
given by
$$\mathrm{AngleInDegrees}(v,w) = \frac{180}{\pi}\mathrm{arccos}\left(\frac{v \cdot w}{\|v\| \cdot \|w\|}\right)$$
Observe that both functions return unitless numbers. So really, degrees and radians aren't units at all; they're not like meters or seconds. They're more like consistent systems of conventions than anything.
If we want to formalize the relationship between these conventions, then $x^\circ$ should be defined as stated in my original answer, as the result of evaluating a function $(-)^\circ : \mathbb{R} \rightarrow \mathbb{R}$ at a (unitless) number $x.$ Explicitly:
$$(-)^\circ : \mathbb{R} \rightarrow \mathbb{R}$$
$$x^\circ = \frac{\pi}{180} \cdot x.$$
It follows that:
$$\mathrm{AngleInRadians}(v,w) = (\mathrm{AngleInDegrees}(v,w))^\circ.$$
Under this convention, statements like $\pi = 180^\circ$ and $\cos(\theta+180^\circ) = -\cos \theta$ are literally true, where $\cos$ is viewed as a mathematical function $\mathbb{R} \rightarrow \mathbb{R}$. So the inputs to $\cos$ are mere numbers; they have no units, and neither do its outputs.
Strictly speaking, there are two functions that are commonly denoted by $\sin(\cdot)$. The mathematical sine function, $\sin: \mathbb R \to \mathbb R,$ has the real numbers as its domain. That is, the function does not take angles to numbers; it takes numbers to numbers.
The domain of the other sine function is angle measurements; an angle measurement consists of a number and the units of measurement of the angle. Just as a single interval of times or a single linear distances can be written in multiple different ways using various numbers with various units, a single angle can be written in multiple different ways with different units. Hence it is correct to write
$$ 180^\circ = \pi\mbox{ rad},$$
where the symbol $^\circ$ indicates units of degrees and rad is the symbol for radians, or (if you are in a context where it is appropriate to use the "other" function named $\sin(\cdot)$) to write
$$ \sin(30^\circ) = \sin\left(\frac\pi6 \mbox{ rad}\right).$$
On the other hand, in a more "pure" mathematical context using the function $\sin: \mathbb R \to \mathbb R,$ strictly speaking we should write
$$\sin\left(\frac\pi6\right) = \frac 12 \neq \sin(30) \approx -0.988.$$
In practice, the tendency to interpret the notation $\sin(30)$ as
$\sin(30^\circ)$ is so strong that if you type sin(30) as input to
Wolfram Alpha (for example) it will return $0.5$ as the answer.
On the other hand if you put =sin(30) in a cell in some widely-used
spreadsheet programs you may be in for a surprise.
One just has to be aware of this potential source of confusion
(identical names for two different functions) and make sure one uses the
correct function in the given context.