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No.When they are on different planes and they do not cross, they are called skew lines, they are not considered parallel. When they ARE parallel, it means that they do not cross and they both lie on ONE plane
Points and lines on the same plane are coplanar.
I sense you're talking about the infinite disk, the hyperbolic disk or the Poincare disk. The limit of the circumference is infinite and a real number and is not actually part of the hyperbolic plane.
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.