I don't know much about these things but I think 1. fails even if $X$ is a point and $Y$ is a line. Let's say also $R$ is a field. Then the category of constructible sheaves on $X$ is just $R$-Vect which is in particular semisimple, but the category of constructible sheaves on $Y$ is not semisimple: if we choose a sheaf on a point $p$ and on the complement $U$ of that point, then there are in general many non-isomorphic choices of a sheaf on $Y$ with given restrictions to $p$ and $U$ which fit in a short exact sequence.
So their derived categories are not equivalent either.
Addendum: Maybe I can say something about how the usual homotopy invariance of cohomology is visible in the 'six functor' formalism. For any space $X$ let $\pi_X$ be the projection to a point, then the cohomology of $X$ (with any coefficients) just corresponds to functor $R \pi_{X\ast} \circ \pi_X^\ast$. If $f \colon X \to Y$ is an arbitrary map, then note that $\pi_Y \circ f = \pi_X$ which implies
$$ R\pi_{Y\ast} \circ Rf_\ast \circ f^\ast \circ \pi_Y^\ast = R \pi_{X\ast} \circ \pi_X^\ast;$$
and now the morphism $\mathbf 1 \to Rf_\ast \circ f^\ast$ coming from the adjunction gives a map $f^\# \colon R \pi_{Y\ast} \circ \pi_Y^\ast \to R \pi_{X\ast} \circ \pi_X^\ast$. Of course evaluating this equation on a choice of coefficients we get the usual map $H^\bullet(Y) \to H^\bullet(X)$. The correct way to express homotopy invariance is now that if $f$ and $g$ are homotopic maps $X \to Y$, then $f^\#$ and $g^\#$ are equal. This can be generalized to the relative situation, when $X$ and $Y$ are spaces over some base space $S$ and we consider the derived pushforward to $S$ instead of to a point, and we take homotopies over $S$.
