Unfortunately, NO!!!  TVs are set at the factory to sell on the showroom floor.  The typical showrooms are very bright.  To overcome this they set the TV very bright so it will catch the consumer's eye.  When the TV is set this bright it often washes out many of the colors of the picture.  To compensate for this they usually adjust the red level higher than the rest because they know most people judge TVs by viewing skin tones on people.  This is often the reason TVs don't look as good when you get the home.

The TV typically won't be as bright, however, the colors are usually more vibrant, the dark scenes show more detail and the TV will last longer.  It will also use less electricity.  A calibrated TV will look better and save you money.  Some TVs have day and night settings so adjustments can be made for both viewing environments.

Sure!  You need some type of instrumentation which usually costs thousands of dollars and the knowledge to use it.

So the grayscale is the levels of the black and white signal being displayed.

This can be a much more complex question than it sounds but I'll try to give you some simple solutions.  In general I'd use HDMI or DVI and there are adapters to convert one to the other.  Sometimes the digital connections have problems with the copy protection information that is being transmitted.  The copy protection called HDCP (High Definition Copy Protection) can cause either loss of picture or will display a picture for only a few seconds.  If this happens use the component connection.