Parallel lines by definition never cross each other.
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In Euclidean geometry, they do not meet.
It is an axiom that parallel lines never meet in Euclidean geometry. Never.However in another kind of geometry (can't remember name) it states that parallel lines will eventually meet.Take a look at this picture in the related link, below.Technically the lines are parallel (in theory, they have imperfections), but due to our perspective parallel lines appear to meet. Note: If they really do meet, then you could drive down the road and eventually there would not be a road, anymore.
In Euclidian geometry, which is the geometry of a plane surface, parallel lines do not intersect because that is the definition of parallel lines. But note that there are other geometrical systems in which parallel lines do intersect, for example if they are drawn on the surface of a sphere. Definition of parallel lines: Lines that always stay the same distance apart and never meet.
In reality they would never meet. But mathematically speaking, they would meet at infinity. Infinity is only a relative term.
No, parallel lines do not meet at a right angle. In theory, parallel lines never meet. In practice, parallel lines on earth could meet at the North Pole and/or the South Pole. Perpendicular lines meet at a right angle.