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Yes. The distribution can be compact (centred tightly around the mean) or spread out. It can have a peak in the centre or two peaks at each end, or other variations.
The importance is that the sum of a large number of independent random variables is always approximately normally distributed as long as each random variable has the same distribution and that distribution has a finite mean and variance. The point is that it DOES NOT matter what the particular distribution is. So whatever distribution you start with, you always end up with normal.
The normal distribution is a theory, which works in practice (with a large enough sample). E.g if you were to plot the height of everyone in the country, you should end up with a normal distribution. Hence it is not usually considered hypothetical, in the same way that, say, imaginary numbers are hypothetical.
Yes. When we refer to the normal distribution, we are referring to a probability distribution. When we specify the equation of a continuous distribution, such as the normal distribution, we refer to the equation as a probability density function.