In the limit where $m_2 \ll m_1$, only the mass of the heavy body matters (along with the semi-major axis of the orbit, of course).
Where that limit does not apply, varying the mass of either body changes the reduced mass:
$$ \mu = \frac{m_1 m_2}{m_1 + m_2} .$$
Since the system acts as if a negligibly massive object was moving in the field of one having the total mass, this does alter the period.
Notice that in the limit above the total mass is approximately $m_1$ and we recover the expected behavior.
Marion and Thorton give the full expression for the period $\tau$ in the form
$$ \tau^2 = \frac{4 \pi}{G} \frac{a^3}{m_1 + m_2} $$
where $a$ is the length of the semi-major axis of the orbit and $G$ is the gravitational constant. It should be obvious that in the limit of a heavy primary this reduces to $\tau^2 = \frac{4 \pi}{G} \frac{a^3}{m_1}$.
Side comment: The rule you recall is the one Kepler found for planets in our Solar System. In this case the mass of the sun dominates in every case. Jupiter is about 0.001 solar masses, so the largest correction in at the tenth of a percent level. Observable, but not at all large.
I'm going to interpret what you're asking as follows: The moon orbits the planet, the planet orbits Sun #1, and Sun #1 forms a binary star system with Sun #2. The planet is close enough to Sun #1 that the gravitational effects from Sun #2 are small.
I believe this is a stable arrangement, particularly if Sun #2 is very far away. It's quite similar to the way that Jupiter's moons are in stable orbits, even though Jupiter and the Sun orbit one another*. In fact I believe planets have been detected that orbit stars in binary systems. (If anyone has a reference, please feel free to edit it in.)
As for what effect this would have on the seasons, that's strongly dependent on the relative brightnesses of the two stars. Since stars can differ wildly in size from one another (see http://what-if.xkcd.com/83/, as well as endless YouTube videos on the subject), it's entirely possible that Sun #2 could be the brightest one in the sky, even though Sun #1 is much closer.
It's also entirely possible that Sun #2 would be just one tiny point of light among all the other stars, perhaps even invisible without a telescope. This is probably the most likely scenario, but it's also a bit boring, so let's assume that Sun #2 is some kind of giant star, so that it's of roughly the same magnitude as Sun #1, as seen from the planet.
There are several possibilities for how the two Suns might move across the sky, which depend on the configurations of the orbits. One possibility is that all four bodies might orbit in roughly the same plane, like a giant version of our Solar system. In this case, there will be a time of year at which the planet and its two Suns are roughly in a line. At this time of year, the Suns would appear close to each other in the sky, rising and setting with one another, just like the famous scene in Star Wars**.
Half a year later, the three bodies would line up again, but this time with the planet in between the two Suns. Then they would appear on opposite sides of the sky, with one rising as the other sets. Thus it would indeed be daylight all the time, as you say.
The light from the two Suns might be different in quality, though. Since Sun #2 is a much bigger star than Sun #1, it's also likely to be hotter. In this case it might seem warmer when Sun #1 is in the sky, but brighter after Sun #2 rises. The light from Sun #1 would also be noticeably redder and the light from Sun #1 noticeably bluer.
If Sun #1 is a very dim star, it might even be possible to observe light from Sun #1 reflected from its surface. In this case it would have phases, just like Earth's moon. The full phase would occur when the two Suns are on opposite sides of the sky, and "New Sun #1" would be when they're in the same part of the sky. (It would be nice to do some calculations to see if that's actually possible. I haven't done that.)
Speaking of phases, the Moon would reflect light from both Suns. It would typically look like two moon phases super-imposed on top of one another. If two stars were different temperatures (thus giving out different coloured light), I imagine this would look really cool.
It's also possible that the planet might orbit Sun #1 in a different plane from the one in which the two stars orbit one another. Let's take the extreme case and say they're at right-angles to one another. I'll assume that the planet's axis of rotation is more or less aligned with its orbit around Sun #1.
Now we have to consider two "years": the time taken for the planet to orbit Sun #1, and the time taken for the two stars to orbit one another, which will necessarily be longer - perhaps up to 100,000 times longer. I'll call this second year the "long year" and the other one the "short year". We can imagine that a short year is something like an Earth year.
If it's the right time in the long year, an interesting thing will happen. The plane in which the planet orbits will be at right-angles to an imaginary line drawn between the two stars. This means that the planet's axis of rotation will point towards Sun #2: it will appear in the same part of the sky thought the whole (short) year, just like Polaris does on Earth. Thus one hemisphere of the planet will have constant light from Sun #2, while the other has only the light from Sun #1, which rises and sets daily.
This situation will change over the long year. A quarter-long-year later (which might be tens of thousands of short years) the Suns will be more or less aligned with the planet's orbital plane, and the situation will be similar to when they all orbit in the same plane, as described above. Another quarter of a long year and the situation is reversed, with the opposite hemisphere being exposed to constant light from Sun #2.
There are other possibilities, depending on the relative alignments of the planet's orbit, its axis of rotation, and the plane of the stars' orbit. Because these won't all exactly line up with one another, the seasons on such a planet would be complicated. On Earth the seasons are due to the tilting of the Earth's axis of rotation relative to its orbital plane. This would likely be in effect on this planet as well, but the location of the two stars in the sky would likely have a bigger effect. Whether these two types of "season" happen in phase or out of phase with one another will vary over the course of the long year.
*Jupiter is just about massive enough that the Sun can be said to orbit it, as well as it orbiting the Sun. The two bodies orbit their common centre of mass, which is inside the Sun but quite near its surface.
**In that scene the two stars appear about the same size in the sky. I think this is unlikely - as a rough guess, I'd say Sun #1 should appear larger, with Sun #2 being smaller in the sky but shining more intensely, like a tiny ultra-bright LED next to a big incandescent bulb. But it does depend on what kind of stars they are, and the Star Wars scene might be possible for all I know. If Sun #1 is large in the sky it might eclipse Sun #2 once per year when the three bodies line up.
Best Answer
It would be possible, but very unlikely, since the orbits wouldn't be stable.
Try to take a look at this visualization of the gravitational potential of a binary star system (from the Wikipedia Roche Lobe entry):
If the planet orbits just one of the stars, its orbit will be inside one of the lobes of the thick-lined figure eight at the bottom part, analogous to a ball rolling around inside one of the "bowls" on the 3D-figure. Such an orbit will be stable, just like the Earth's around the sun (bar perturbations from other planets, but let's leave them out for now), and there will be many different orbital energies for which this is true.
The same goes for an orbit around both stars: the planet will have many different energy levels at which it would simply experience the two stars' gravity combined as the gravity of one single body (and in which case the figure wouldn't apply, since it would be practically unaffected by the two stars orbiting each other).
In order to orbit in a figure eight, you have to imagine that the ball has to roll across the ridge between the two indentations in the 3D part of the figure. It is clear that this is possible, but also intuitively clear that this would only be possible for a narrow range of orbital energies (a little less and it would go into one of the holes, a little more and it would simply just orbit them both), and that it would not be a stable orbit. The ball would have to roll in an orbit where it exactly passes the central saddle point at the ridge (L1) in order to stay stable, the tiniest little imperfection will get it perturbed even further away from its ideal trajectory.
Your 5-body system could possibly be timed in such a way that it would work, but it would suffer the same fundamental flaw, and as far as I can see, it would also introduce even more sources of instability into the system.
This is, by the way, the gravitational potential in the rotated coordinate system, and you can see from the symmetry of the system that the coreolis preference you mention is not present. A simple symmetry argument should convince you of the same, though: Assume the system is rotating clockwise. This should allegedly give you a preference for one of the stars. But if you now let the system continue, while you rotate yourself 180 degrees up/down, it will now be rotating counter clockwise, which should give a coreolis preference for the other star, which of course cannot be the case, since there is no preferred up/down direction in a system like this.