Are the Northern Hemisphere’s autumnal equinox and the Southern Hemisphere’s vernal equinox at precisely the same time? Or is there some sort of anomaly, due to the earth’s irregularities?

It’s a worldwide single event that occurs twice a year.

The equinox occurs at the instant that the subsolar point (the single point on Earth at which the centre of the sun can be observed directly overhead) is on the Equator. Since this is a single event, there is only one equinox for the whole earth.

(The solstices occur at the instant the subsolar point is the furthest north (at the tropic of Cancer) or furthest south (at the tropic of Capricorn) it can be.)

And because of this, the sun will rise due-east and set due-west no matter where you are on earth, dividing the day and night into, more or less, equal amounts of daylight and darkness. (of course your latitude will determine the arc the sun traces in the sky: the sun will reach zenith at noon on the equator, and at the poles, it’ll appear to revolve around the entire horizon for an even 24 hour twilight).

Because the equinox is measured based on the centre of the sun, on the day of the equinox daylight lasts longer than twelve hours because the limb of the sun rises earlier and sets later than the centre of the sun does.

The day on which there are equal periods of day and night (or at least the day when they’re closest to equal) is called the equilux. The date of the equilux varies from place to place.

And, of course, the atmosphere refracts the light during dusk and dawn, so, even if the sun is truly below the horizon, it lenses the image of the sun above it, adding more daylight (by several minutes) than we’d actually get if there was no atmosphere.