In this hypothetical solar system the one habitable planet in question would be at center of two sun stars, one similar to our sun and the other a red dwarf. What kind of effect would this have on the planet's day/night cycle, weather, life? Also, could humans adapt to such an environment?
Yes, but not in the way you imagined.
Vincent mentioned a Worldbuilding question which directly addresses the possibility of a figure-8 orbit. No offense intended, I don't think any of the answers there properly addressed the question - although Vincent may have decided against addressing it because a Physics question had already covered it, which he mentioned in his answer, and it had a fairly comprehensive answer.
Quoting Thriveth in his/her answer there,
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.
So such an orbit is possible, but very unlikely, and pretty unstable.
That doesn't mean that a planet couldn't follow a circumbinary orbit around both stars. Indeed, many planets have been found with such orbits. And it means that your 40-hour day and 3-hour night is possible.
Look at the Moon on the right side of this animation (I couldn't separate the two):
Image courtesy of Wikipedia user Stigmatella aurantiaca under the Creative Commons Attribution-Share Alike 3.0 Unported license.
This Moon (incorrectly; our Moon is like the one on the left) doesn't rotate from the perspective of a fixed observer above the Earth. Still, the face pointed towards the Earth is constantly changing.
Now change the Earth to a pair of close binary stars, and the Moon to our planet. In fact, make the orbit really eccentric, so it swings out to the far reaches of the system. Because of Kepler's second law, it travels faster near the stars. This means that for one half of the planet, the night will last for 3 hours and the day will last for 40 hours; for the other half of the planet, the night will last for 40 hours and the day will last for 3 hours. To my knowledge, there is no configuration that will let you have the same length of night for both sides and still have the 40/3 scenario.
With a 8 shaped orbit: yes but the length of day/night will not be constant. When the planet is between both stars, the night is almost nonexistent. When the planet faces both stars in the same direction, the night last about the same length that on Earth depending on the axial tilt.
Seasons will be determined by the position of the planet in relation to the stars and also amplified by the planet axial tilt if any. Between both stars is the hottest. Close to the red star, at the exterior of the orbit is the coldest. But it's not necessarily cold enough to be considered winter. It depend on the distance with the stars.
Below are a few solutions to the question :
The simpler and realistic one first : You orbit a small planet very close to its sun, and you spin it naturally.
Because your planet is much smaller than the sun and you are close, rays coming from the side of the sun can reach a significant fraction of the opposite side of your planet.
To see it : Imagine a big circle and a small circle, and draw the two tangents to those two circles.
Geometrically any X with 50% < X < 100% of the surface of your planet can be illuminated depending on the distance orbit and the planet and star sizes. In your setup you wish for a X of ~ 90%.
Note that in 3 dimensions, a significant fraction of your planet which will receive light 100% of the time (when |latitude| > Y ). But at the equator you will have your ~90% of day night imbalance.
Another solution : add atmosphere effects to a planet : If you have an atmosphere reflective and diffuse just the right way, light can bounce a few times in it to reach the back side of the planet and to lit it. (By the way on earth L.W. radio waves do just that with our atmosphere.)
Another solution : Donut shaped planet :) will self-shadowed and indirect light on some points and by settings the 2 radius of the torus correctly you can specify the day/night imbalance. Play with a ray tracer to convince yourself.
Here is yet another solution : Solar winds which by interacting with the magnetic field of the planet makes polar lights which illuminate so strongly the would be night.
Yet another solution using general relativity : The difference of gravity between the closer to the sun side and the farther from the sun side means that times goes slower on the lit side, so the day are longer, but I would bet that tidal forces would make such scenario unrealistic :( :)...