parallel planes
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Yes. If two planes are not coincident (the same plane) and are not parallel, then they intersect in one straight line.
The angle between two planes when the planes intersect at a point is the acute angle fixed by the normal vectors of the planes.
No, horizontal planes run parallel to each other, so they do not intersect, but two vertical planes can intersect. Imagine the pages of a books as several planes. When you stand the book up, they are all vertical, but they all intersect at the book spine.
No. Two planes may be parallel and so may not intersect. Also, any line is the intersection of infinitely many planes, not just two.
There are many options. Amongst them: An ellipsoid (including a sphere) intersected by two planes which do not, themselves intersect within the shape; A paraboloid intersected by two planes which do not, themselves intersect within the shape; A cone intersected by two planes which do not, themselves intersect within the shape; A semi-hyperboloid intersected by two planes which do not, themselves intersect within the shape; A toroid (doughnut) intersected by two planes which do not, themselves intersect within the shape; An elliptic prism intersected by two planes which do not, themselves intersect within the shape. The last of these would include a cylinder.