infinitely long
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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.
Lines which are parallel. All other lines on the same plane eventually intersect.
An arc-hyperbolic function is an inverse hyperbolic function.