Are the stars seen at the equator different throughout the year? If you live on the equator and look up at the night's sky at, say, midnight, would you see a completely different sky on the 1st of January than you would on the 1st of July?

I assume you would as the night side would be facing away from the sun by 180 degrees and necessarily looking outward from the solar system 180 degrees from each of the two points at those times.

Is that correct?
Why?
Yes, the constellations that are visible at midnight on Jan 1 are different than those visible at midnight on July 1 (6 months later). The website Stelvision has a star chart that you can customize for date, time, and location. This is due to the Earth's revolution around the Sun and the direction that an observer is facing at midnight.

From a given location, a given star or constellation rises, transits (when an object is due south), and sets 2 hours earlier each month. 

In generally, the above is true for all latitudes. Of course, stars/constellations that are circumpolar never rise and set, and from the Earth's pole all objects are circumpolar. But even in this case, if you would pick a direction to look at midnight on Jan 1, those constellations would be behind you (180 degrees away) at midnight on July 1 (assuming you could see them in the daylight on one of those dates ;-).