triangle, square, retractable, polygon, circle, eclipse in 2d. pyramid, cube, box, what for polygon, sphere, ellipsoid in 3d. what tesseract in 4d?...

Question

What is a corresponding shape's name in 3d, 4d and is 5d have some name for it:

2d: triangle, square, retractable, polygon, circle, eclipse
3d: pyramid, cube, box, what for polygon, sphere, ellipsoid 
4d: tesseract, what are rest?
5d: is some name here also
6d:
..
..
nd:

is something infinite dimension that exists?

Answer 1

Name of the polygon in $3$-dimension are:
$~~~~~~~~~~~~~$tetrahedron, Hexahedron, octahedron, Dodecahedron, Icosahedron, polyhedron etc.
List of regular polytopes and compounds

The prefix "hyper-" is usually used to refer to the four- (and higher-) dimensional analogs of three-dimensional objects, e.g., hypercube, hyperplane, hypersphere. n-dimensional polyhedra are called polytopes.

The four-dimensional cases of general $n$-dimensional objects are often given special names, such as
$~~~~~~~~~~~~~$glome, tesseract, pentatope, polychoron, hyperplane, 16-cell, facet, content etc.

For more details you may see the following references:
Four-Dimensional Geometry
Viewing Four-dimensional Objects In Three Dimensions

General $n$-dimensional objects:
$~~~~~~~~~~~~~$hypersphere, hypercube, simplex, polytope, hyperplane, cross polytope, facet, content etc.

For more details you may see the following references:
Higher Dimensional Polytopes
Shapes, Fractals, Time &, the Dimensions they Belong to
Exotic Objects from Higher Dimensions