If they day is shorter in the winter does that mean the sun moves faster across the sky?

The Sun doesn't move any faster, but it does spend less time above our horizon in the winter (I live in the northern hemisphere, and I'm assuming you do, too).

On the first day of spring and the first day of fall, the Sun rises exactly east, reaches it's highest point in the sky when it is south (for most people it never goes directly overhead), and then sets directly west. Let's say for sake of argument that you live at a latitude of 40 degrees north of the equator. Then the Sun on these days will get 50 degrees above the horizon at its highest, when it is due south. Also on these days there will be 12 hours of daylight.

On the first day of summer, the Sun actually rises in the northeast, and not directly east. So the Sun will be due east sometime in the morning. It will still reach its highest point in the sky when it is due south - the Sun will be 73.5 degrees above the horizon (50 + 23.5, the tilt of Earth's axis). It will set in the northwest. So on the first day of Summer the path of the Sun above the horizon is longer than it was on the first day of spring or fall, and we have more than 12 hours of daylight.

Likewise, on the first day of winter the Sun rises south of east, reaches a height of only 26.5 degrees above the southern horizon, and sets south of west. So the Sun's path is shorter, and we have less than 12 hours of daylight.

Sundials will still be accurate - of course sundials can't be corrected for daylight savings time, nor do they know what time zone you are in!

#1. No, the sun does not move faster. Caitlin has been closest thus far. The tilt of the earth means that the sun is travelling a shorter distance from horizon to horizon. It doesn't go from due east to due west (except on the equinoxes). Other times it makes a longer or shorter arc across the sky. This can best be observed in the far north (or far south) above the arctic circle. At the summer solstice, the sun will travel in a circle in the sky, never going below the horizon.

#2. Sundials are always inaccurate, compared to clocks. They measure solar time, which is not the same as calendar time. However, they will accurately denote solar noon, regardless of the season.

No. Keep in mind that the days are only shorter in the Northern Hemisphere ( the Southern Hemisphere is having longer days now) and the sun doesn't really move. The sun would only appear to move faster across the sky if the Earth began rotating on its axis faster. The days are shorter because the part of Earth we are on is turned away from the sun this time of year.

There is more than one sense of the word day. Day could mean a period of roughly 24 hours. That doesn't change as it is dependent upon the Earth's rotation which doesn't really change that much about its axis. The other sense of the word day refers to the time during a 24 hour period in which there is daylight or rather light from the sun is directly visible. That period of time changes with the season due to the angle of the Earth's axis with respect to the direction of sunlight. The Earth's axis is tilted. At one season, more of the Earth's surface with respect to the hemisphere is available to direct exposure to sunlight. During winter in the Northern hemisphere, we are physically closer to the sun. But, since the sun's rays are so indirect, we are also colder.

The sun starts setting later a couple weeks or more before the solstice, but the days are still getting shorter because the sun is also rising later. The latest sunrise will happen in January sometime.

the day is shorter in the northern hemisphere because of the angle that Earth rotates on its axis... the sun doesnt move faster, and posisbly sundials may be inaccurate

LOL You have to be kidding.

The earth in relationship to the sun is at a different angle during winter. This dispurses the suns rays slightly. Take a flaslight and point it straight down... now angle the beam on the floor...see how it's less concentrated? Thats what makes it colder... and since we are at that different angle...we have less exposure to the sun

this would not make sun dials inaccurate, the sun does not move faster across the sky, the angle of the earth in relation to the sun focuses the sun on the soutern hemisphere of the earth therefore they get more sunlight than we do when it is winter...

no it has to do with the tilt of the earth during winter months we are tilted away from the sun thus resulting in shorter days

No , it never moves faster. We manipulate our clocks to have the 1 hour gain or loss of time.