Put P inside Q! polygons/polyhedra We have two Polygons/Polyhedra P and Q.
Does there exist a polynomial time algorithm to decide if we can put P (using translation and rotation) inside Q or not?
First think about the case of which P and Q are convex and the input is the list of vertices, edges and faces of P,Q.
 A: To make sense of this question one has to say first what you mean by "copy of $P$".
For example one could image to allow translations, rigid motions and/or dilations.
The answer also depends on how the the (convex) polytopes are given.
There is some literature on this topic, a starting point could be:

Peter Gritzmann and Victor Klee. On the complexity of some basic
  problems in computational convexity: I. containment problems. Discrete
  Mathematics, 136(1):129–174, 1994.

From this paper:

This Theorem does not consider a "copy of $P$" but only $P$ itself, other cases lead to cases that don't have a polynomial time algorithm.
For the polygon-case a good reference might be:

Bernard Chazelle. The polygon containment problem. In Franco P.
  Preparata, editor, Advances in Computing Research I, pages 1–33. JAI
  Press, 1983.


Edit: after your clarification let my restate the question:

Given two $\mathcal{V}$-polytopes $P$ and $Q$, is there a polynomial
  time algorithm to find a $\mathcal{V}$-polytope $P'$ such that
  $P'\subset Q$ and $P'$ can be transformed into $P$ by rigid motions?

The answer to this question is: No
The relevant section of the first reference above is 
Section 7. Optimal containment under similarity and related questions.
As mentioned there, this question is already difficult if $Q$ is a cube. Note the they look at the closely related question of finding a copy of $Q$ that contains $P$. (Instead of "contains $P$" we can simply ask for "covers the vertices of $P$", if convexity is assumed)
A: There is quite a bit of literature specifically on polygon containment,
much of deriving from robotics applications. The paper below solves the containment problem,
allowing translations and rotations, with a polynomial-time algorithm (roughly $O(n^6)$).

Avnaim, Francis, and Jean Daniel Boissonnat. "Polygon placement under translation and rotation." Informatique theorique et applications. 23(1). 1989. p.5-28.

Here is a snapshot from their introduction. The two polygons,
named $E$ and $I$, have $n$ and $m$ edges respectively. This Intro reviews
related, prior work:

 
 
 


Their algorithm extends to handle the case of polyhedra in $\mathbb{R}^3$,
but for translation only (no rotation).
