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No. Skew lines do not intersect
Skew lines never intersect. If two lines intersect, then they are known as "intersecting lines", not skew lines.
No, skew lines cannot be in the same plane, since they do not have a point on common. Two lines intersect if they lie in a common plane, and by definition, these intersecting lines are not skew lines.
No. Two lines can include the same point only if they intersect.
If the 2 lines lie in the same plane, and they are not parallel, then they will intersect at some point. If the 2 lines are skew lines, then they are not in the same plane, and they will not intersect (but they are Not Parallel)