You have a lot of good questions! It seems like you already have learned a lot about how the Moon, Earth, and Sun move together to produce the seasons, so understanding more how the Moon works will be a snap.

To answer your first question, the length of time the moon is visible does change from season to season, just like it does for the sun. In fact, the amount of time the moon is visible is flipped compared to the sun, so when the sun is visible for a long time, the moon is visible for a short time, and when the moon is visible for a long time, the sun is visible for a short time.

The reason the moon varies at all is for the same reason the sun varies, the tilt of the earth. As you probably have already learned, when the earth is tilted away from the sun, the sun will appear close to the horizon, and when the earth is tilted towards the sun, then the sun will appear high above the horizon.

If you think about the moon, you might realize that the moon is brightest and most visible when it is on the opposite side of the earth from the sun because that is the side of the earth in nighttime. If the sun and moon are on opposite sides, then when the earth is tilted toward the sun, then it must be tilted away from the moon! So just as the sun wasn't visible for a long time when the earth was tilted away from it, the moon won't be visible for long when the earth is tilted away from it. This is why the moon varies in the amount of time visible opposite to the sun.

Now onto your second question. You are right that the earth and the moon are not orbiting in the same plane, and I believe that a picture will greatly help explain how their orbits differ.

A diagram of the orbits is available at diagram of the orbits .

It will helpful if you can look at the picture alongside reading my explanation.

What the picture shows is that the earth lies in the Ecliptic Plane, but it is tilted 23.5 degrees with respect to the Ecliptic Plane. That's the tilt that explains the changes in the amount of time the moon and Sun are visible. The diagram also shows that the moon's orbital plane lies tilted at 5.15 degrees with respect to the Ecliptic Plane, and the Moon is tilted at an angle of 6.18 degrees with respect to its orbital plane.

No - the length of time that the moon is up does not depend on the season. The reason why the earth has different lengths of day is because its axial tilt is fixed and not perpendicular to its orbit. As a result, different parts of the earth get different amounts of sunlight during different times in the orbit, creating seasons.

The moon's "year" so-to-speak, is 28 days - the length of time that it takes the moon to go one complete revolution around the earth. This would mean that if the moon's orbit is different from the earth's axial tilt (which it inevitably must be, else eclipses would be much more common), then the moon's visibility would vary over the course of the lunar orbital cycle. However, as I said, the lunar orbital cycle is only 28 days, so over a dozen of them fit in a single year.