There's a neat numberphile video on the subject! [1]

[1] https://www.youtube.com/watch?v=2s4TqVAbfz4

Yes. The "n-tetrahedron" is usually called the simplex, and the "n-octahedron" is usually called the cross-polytope.

And the "n-cube" is the "measure-polytope" because you can use squares to measure 2-space, cubes to measure 3-space, hypercubes for 4-space and so on...