No.
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The answer depends on the inclination of the intersecting plane. The cross section can be a point, a triangle, a rectangle, a pentagon or a hexagon.
There is no specific name. For example, if you number the sides of a regular hexagon sequentially from 1 to 6, then sides 1 and 3 are not parallel but there is no specific name for that pair. In the context of the hexagon they do not meet - even if they do so way outside the hexagon. If they do meet up in the context of the shape, then they will be adjacent or intersecting sides.
these are intersecting lines.
Yes, squares do have intersecting lines. Infact they have 2 diagnol intersecting lines.
They are simply called intersecting arcs.