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The linear discrete time interval is used in the interpretation of continuous time and discrete valued: Quantized signal.
A variable defined on a continuous interval as opposed to one that can take only discrete values.
A 1-dimensional interval (a, b) is continuous if for any k in (0, 1) the point a + k*(b-a) = a*(1-k) + k*b is also in the interval. This is equivalent to the statement that every point on the line joining a and b is in the interval. The above can be extended to more dimensions analogously.
No. If the variable is continuous, for example, height or mass of something, or time interval, then the set of possible outcomes is infinite.
Yes, it is a Continuous variable measured along an equidistant scale.