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Which solar system objects have the right combination of low gravity and fast spinning rate, that an object tall enough to be a space elevator could be built on with modern construction tech?
By "modern construction tech", I'm referring to readily available reinforced concrete of the sort they used to build the Burj Khalifa, not carbon nanotubes or any other 'theoretical' building materials.
Are there any other considerations, such as the presence of moons, that need to be accounted for?
Here's how you can calculate this for any spinning body.
First, you need its stationary orbital radius
$$r_{stationary} = \sqrt[^3]{\frac{\mu}{\omega}}$$
Secondly, we need to find how many Yuris (m²/s²) the space elevator is.
To obtain that, we can first look at what acceleration is acting on the elevator in a co-rotating frame of reference at some radius.
$$a(r) = \omega ^2 r - \frac{\mu}{r^2}$$
Integrating that over distance, we get the specific strength requirements:
$$\int_{r_{surface}}^{r_{stationary}} a(r) dr = \frac{\omega^2 r_{stationary}^2}{2} + \frac{\mu}{r_{stationary}} - \frac{\omega^2 r_{surface}^2}{2} - \frac{\mu}{r_{surface}}$$
Or somewhat more compact:
$$Y_{elevator} = \frac{\omega^2}{2} \left(r_{stationary}^2 - r_{surface}^2\right) + \frac{\mu}{r_{stationary}} - \frac{\mu}{r_{surface}}$$
This can be used to calculate the cross section ratio "taper ratio" between the upper part and the lower part of the elevator:
$$A = e^{\rho \cdot Y_{elevator} /T_s}$$
Where $\rho$ is the density of your material of choice, and $T_s$ is its tensile strength
As always with exponential equations, they grow very quickly. A cross section ratio of 10 is probably okay, while 1000 is out. And that's a difference in material strength of only a factor of 3.
Are there any other considerations, such as the presence of moons, that need to be accounted for?
Very few solar system bodies have moons below their stationary orbital altitudes that can get in the way. The Mars moon Phobos is one of the few examples.
Another consideration to make is that almost all moons in the solar system are tidally locked, and thus don't really have a stationary altitude. In those cases, you can build an elevator to their L1 and L2 Lagrangian points instead. The equation is a bit longer (not too hard! just do the integral above with another mass added), however you can use the following approximation, which is only a tiny bit too pessimistic:
$$Y_{Lpoint} = \frac{\mu}{r_{surface}}$$
Solar system space elevator requirements: (MY = megayuris)
