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To intersect, their slopes have to be different. The y-intercept can be anything.
If parallel lines had the same y intercept then they would intersect each other at that point (0, c). Parallel lines don't intersect at all so they cannot have the same y intercept. The only exception would be if both lines were the same.
Knowing that they have the same y-intercept, and knowing nothing else, the only thing you can say about the two lines is that they have the same y-intercept. That fact doesn't tell you anything else about them.
Yes, but only of the line is vertical. (It would then be written as x = a, where a is a constant. For example, the line x = 8 does not have a y-intercept. All lines in the form y = mx+b will have a y-intercept.)
No, vertical lines have an undefined slope.