2 strike out a flawed remark

A personal preference, undoubtedly, but I find that a differential geometry point of view can really help with the questions you're asking.

Dimension: Any semialgebraic set can be decomposed (non-uniquely) as a union of smooth manifolds (stratification), and you can always define the dimension as the maximal dimension that occurs in your decomposition. It does not depend of the decomposition. Of course, there are other ways of defining dimension, especially for varieties. But note that your set can have smooth points of varying dimensions (e.g. Whitney's umbrella or Cartan's umbrella).

Bézout-type results: There are many in real algebraic geometry, all made the more complicated by the fact that real roots don't always exist, of course. In your case, the result would follow from the fact that there are effective bounds on the number of connected components of intersections of algebraic varieties (and other semi-algebraic sets) in terms of the defining equations. The best bounds come from the critical point method, which is an efficient algorithm to produce one point per connected component, based on finding critical points of fairly simple Morse functions (e.g. generic projections or distance from a generic point). The bounds come from the Bézout inequality applied to polynomial systems defining those critical points. So a couple of derivatives of your polynomials will show up, and having a million points is complete overkill in your case, provided that you can guarantee that the intersection must be 0-dimensional when $P$ is not contained in $Q$ (which should be the case here). (oops! I was thinking of the wrong dimension!)

Added later: See section 12.6 of the book by Basu-Pollack-Roy for an exposition of the critical method. It is downloadable free of charge at the link.

1

A personal preference, undoubtedly, but I find that a differential geometry point of view can really help with the questions you're asking.

Dimension: Any semialgebraic set can be decomposed (non-uniquely) as a union of smooth manifolds (stratification), and you can always define the dimension as the maximal dimension that occurs in your decomposition. It does not depend of the decomposition. Of course, there are other ways of defining dimension, especially for varieties. But note that your set can have smooth points of varying dimensions (e.g. Whitney's umbrella or Cartan's umbrella).

Bézout-type results: There are many in real algebraic geometry, all made the more complicated by the fact that real roots don't always exist, of course. In your case, the result would follow from the fact that there are effective bounds on the number of connected components of intersections of algebraic varieties (and other semi-algebraic sets) in terms of the defining equations. The best bounds come from the critical point method, which is an efficient algorithm to produce one point per connected component, based on finding critical points of fairly simple Morse functions (e.g. generic projections or distance from a generic point). The bounds come from the Bézout inequality applied to polynomial systems defining those critical points. So a couple of derivatives of your polynomials will show up, and having a million points is complete overkill in your case, provided that you can guarantee that the intersection must be 0-dimensional when $P$ is not contained in $Q$ (which should be the case here).

Added later: See section 12.6 of the book by Basu-Pollack-Roy for an exposition of the critical method. It is downloadable free of charge at the link.