An equilateral quadrilateral is not a well defined shape - it could be a rhombus with one angle able to take ANY value. In the same way, an equilateral dodecagon is not a well defined shape and so it is impossible to give a sensible answer to the question.
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equilateral triangles
equilateral triangle ;)
Yes in equilateral triangle.
Every isosceles or equilateral triangle.
In general, they are not. In an isosceles triangle, the perpendicular bisector of the base is the same as the bisector of the angle opposite the base. But the other two perp bisectors are not the same as the angle bisectors. Only in an equilateral triangle is each perp bisector the same as the angle bisector of the angle opposite.