bell shaped
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A bell shaped probability distribution curve is NOT necessarily a normal distribution.
Not necessarily.
The Normal curve is a graph of the probability density function of the standard normal distribution and, as is the case with any continuous random variable (RV), the probability that the RV takes a value in a given range is given by the integral of the function between the two limits. In other words, it is the area under the curve between those two values.
Yes. The total area under any probability distribution curve is always the probability of all possible outcomes - which is 1.
the variance is infinitely large and in the extreme case the probability distribution curve will simply be a horizontal line