Depends a lot on your location, but the "star" you are referring to could be the planet Venus or the star Capella
It is visible, but not obviously as a planet. It is millions of miles away, and appears as a very bright star near the horizon, shortly before dawn or shortly after sunset. For some of the year, it is invisible because it appears too close to the Sun.
Sorry; you JUST missed it! For the past few months, Venus has been the "Morning Star" rising before the Sun. But Venus is too close to the Sun to be visible now (December 27, 2009) and is about to go behind the Sun. By the middle of February, 2010, Venus will become the "Evening Star" setting just after the Sun does. By the first week in March, 2010, it should be easily visible in the west right after sunset.
The orbits of the Earth and Mercury are not in synchrony and so there is not a particular time of the year when Mercury can be seen. Obviously, it will not be visible when it is behind the Sun. Also, it will not be visible when it is between the Sun and the Earth because the light from the Sun will obscure it. When Mercury and the Earth form nearly a right angle, with the Sun at the vertex, then Mercury will be visible at around sunrise or sunset - depending on which side of the sun it is.
Originally, the ancients thought that Venus was two different stars. The didn't realise that it was not a star, but one planet which reflects the sun's light. Whe it is visible in the morning it reaches its maximum brightness shortly before sunrise, and when it is visible in the evening it reaches its maximum brightness just after sunset - which is why it got the names Morning Star and Evening Star.
Dark spots on the sun that are visible with the naked eye at sunrise/sunset are sunspots, which are cold areas caused by the sun's magnetic field.
The Sun is usually not seen after sunset, which is why it is called sunset. There is still light visible from the sun. However ther are certain atmospheric conditions that may make it appear as though one sees the sun after setting, but this would be concidered a reflection of sorts.
That actually not a star but the planet Venus.You could also be referring to the planet Mercury. It can be seen just before Sunrise and just after Sunset. As mercury is nearest planet to the Sun, it is always seen very close to the Sun in the Sky. We can't see it in the morning because the is too bright and at night it is below the horizon. Hence, it is visible only for a few moments just before Sunrise and just after Sunset very near the horizon.
The red giant star Betelgeuse is the bright reddish star at the "shoulder" of Orion, the Hunter. Orion is one of the most distinctive constellations in the winter sky, and is easily visible in the winter and early spring. In this season, Orion is setting as the Sun goes down, and it will not be especially visible until the late fall again, when it will rise at sunset.
Proxima Centauri is the closest star to our sun. It's a red dwarf, and is not visible to the naked eye.
No, it is not possible to see the sunrise and sunset at the same place in a single day. The sun rises in the east and sets in the west due to the Earth's rotation. However, during certain times of the year in polar regions, there may be phenomena like the midnight sun where the sun is visible at midnight, but this is not simultaneous with sunset.
No. The sun is a fairly average sized star out of billions of stars.
It is only the brightest visible star because of its proximity. It is a very ordinary one.