Taking the mystery out of geopotential coefficients Geopotential coefficients are usually notated Cnm and Snm (for example, see this IERS Tech Note).
My questions (in the order I care about them) are,

*

*What are the constraints on n and m?  Is n always greater than m?  Can either of them be zero or negative?

*Is it correct to say that n is the "degree" and m is the "order"?

*Sometimes I see the subscripts reversed (Cmn and Smn).  Is that simply a typo?

*Why were the letters C and S chosen?  (Maybe S stands for Stokes or sectoral?  No idea what C might stand for.)

 A: 
What are the constraints on n and m? Is n always greater than m?

The degree ($n$) is always greater than or equal to the order ($m$).

Can either of them be zero or negative?

I don't know whether you're asking about the values of $n$ and $m$ or the values of the coefficients $\bar C_{n,m}$ and $\bar S_{n,m}$. $n$ and $m$ are always non-negative. The coefficients can be positive, negative, or zero. For example, the standard approach is to use the Earth's center of mass as the central point, which makes the first order terms identically zero. The zeroth degree sine coefficients are typically represented as zero. (Any finite value would do for $\bar S_{n,0}$ because the contribution is zero due to multiplying by $\sin(0*\lambda) \equiv 0$.)
Another example is $\bar C_{2,0}$, which is negative. This reflects the fact that the Earth is an oblate spheroid, more or less.

Is it correct to say that n is the "degree" and m is the "order"?

That is the typical meaning. However, ...

Sometimes I see the subscripts reversed (Cmn and Smn). Is that simply a typo?

A few places use $m$ for degree and $n$ for order. A few places reverse the order of the published coefficient values. One always needs to read the literature associated with a gravity model.

Why were the letters C and S chosen?

Cosine and sine. The contributions to the expansion for a particular degree and order are $\bar C_{n,m}\cos(m\lambda)+\bar S_{n,m}\sin(m\lambda)$.
