In the northern hemisphere, the sun is generally in the southern half of the sky. Then, at noon, a shadow cast by anything in New York will point approximately North.
there would be no solar shadow However there might be a lunar one
the shadow would be longer in the winter i couldn't tell you why, though
Yes it does.
northwest
I believe you mean the lightest part of Earth's shadow casted upon the moon and the answer to that would be: The lightest part of Earth's shadow upon the moon is called the penumbra. In fact any shadow is divided between that totally blocked (umbra) and partially blocked by the light source (penumbra) so the Penumbra can indeed be the moons partial shadow on the earth during a partial or total solar eclipse
I would need more information to provide a direction from the starting point to the ending point. Factors such as compass directions, landmarks, or a map would be helpful in determining the direction.
Noon (standard time).
It would be elongated and opposite to the sun. In which direction it will face depends on where you are standing on Earth.
Just after sunrise, or just before sunset.
To see Earth's shadow on Earth, you would need to be at a high vantage point during a time when the sun is low on the horizon, such as at sunrise or sunset. As the sun sets or rises, Earth's shadow is cast onto the atmosphere opposite the direction of the sun, creating a dark band visible in the sky.
That depends where you are on the planet. Your shadow would be longest if you were standing on a rotational pole at the time of equinox. At this point your shadow would be infinity long as it would be "projected" into space. However as a rule of thumb one's shadow is longest when the sun is rising or setting but high enough to cast your shadow.
In the UK, the shadow is likely to fall (point) towards West-north-west.
Thrust would point in the direction of travel. Drag would counter thrust, so generally in a direction 180 degrees from the thrust vector's direction. The lift vector would point in the direction (generally) away from the center of the earth. The gravity vector would point toward the center of the earth.
The exact direction will depend on your geographical location. The general direction will be towards the east, since the Sun will be towards the west. In general, the shadow is opposite to the Sun.
That would depend on your starting point.
Your shadow will be in front of you because shadows will always be in the opposite direction from light sources.
northwest