Do stars in a binary star system fall along the ecliptic?

Most likely, yes.

I wrote an answer here discussion how binary stars are born. The relevant part here is that there are several main theories for how binary stars form:

1. Capture: In an interaction between three stars, two become bound together while the third one is ejected away. This is unlikely to be the major formation mechanism for most binary systems.
2. Separate nuclei: Two stars formed very close to one another, close enough to start orbiting each other. This is also very unlikely to happen.
3. Fission: A protostar collapsing from a gas cloud to a normal star splits in two after some instability forms. Modeling so far rules this mechanism out.
4. Fragmentation: Fragmentation is to some extent like fission, except it involves the gas cloud itself splitting before a protostar forms. This is generally viewed as the accepted way most binary star systems form.

Depending on which hypothesis is true, circumbinary protoplanetary disks should form in the same plane as the orbits of the stars, meaning that the ecliptic would match up (see these notes for more details). There might be some variation - maybe less than 10 degrees - but not much.

In the case of planets orbiting only one of the stars, things are different. There are plenty of cases of single-star disk misalignment in young binary systems. For instance, Jensen & Akeson (2014) looked at the young system HK Tauri and measured a misalignment of 60 to 68 degrees between the protoplanetary disks around each star, meaning that at least one is rotated quite far outside the orbital plane of the stars.

Let me address your specific points.

I believe at some timepoints, one sun is visible in the sky (a sun-on-sun eclipse).

This will be possible, yes, but bear in mind that one star may not fully cover the other, so you'd almost certainly be able to see parts of both.

At other times, they could be discerned as two bodies, and the distance between (and rate of change) would depend on how quickly they are orbiting one another.

This is correct.

Intriguingly, there might be two sunsets (again, similar to Luke's view) - and I think if so it would have to do with how far apart they are. I think the rotation of the planet allows two sunsets for two suns lying in an ecliptic with a planet. Is my space perception way off, here?

Yes, this is also possible. For a planet orbiting both stars, assuming Earth-like rotation, there will always be instants where each star sets separately, as viewed from a certain point on the planet.

posted about 7 years ago

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