I am not an expert in the field, so my question might be rather standard. Let $X$ be a compact metric space. Assume that $X=\cup_{i=1}^NS_i$ is a finite disjoint union of locally closed topological submanifolds which will be called strata. Assume in addition that the closure of each stratum is a union of several other strata.
Let $D^b_{c}(Sh(X))$ denote the bounded derived category of sheaves of vector spaces over some fixed field with constructible cohomology with respect to the above fixed stratification. In other words, objects of $D^b_{c}(Sh(X))$ have precisely such cohomology sheaves that their restriction to each $S_i$ is a local system.
QUESTION. Under what extra conditions on the stratification, the category $D^b_{c}(Sh(X))$ has the following properties:
(1) Let $p\colon X\to pt$ be the morphism to the point. Then for any $\mathcal{F}\in D^b_{c}(Sh(X))$ the push-forward $p_*\mathcal{F}\in D(Sh(pt))$ has finite dimensional cohomology.
(2) For any locally closed subset $i\colon Y\hookrightarrow X$ which is a union of some of the strata and for any $\mathcal{G}\in D^b_c(Sh(Y))$ one has $i_*(\mathcal{G})\in D^b_{c}(Sh(X))$.
(3) For any $\mathcal{F}_1,\mathcal{F}_2\in D^b_c(Sh(X))$ the inner $Hom$ $$\underline{Hom}(\mathcal{F}_1,\mathcal{F}_2)\in D^b_c(Sh(X)).$$
(4) The dualizing complex belongs to $D^b_c(Sh(X))$, and hence (by (3)) the Verdier duality preserves $D^b_c(Sh(X))$.
Remarks. (1) The standard situation of real subanalytic strata is not sufficient for my purposes.
(2) I have heard that often one assumes some local normal triviality condition. However I have not found a reference with a proof of all the above properties for such stratification. Moreover in the literature I saw it was assumed that there are no strata of codimension 1; this is not satisfied in my case.
(3) A reference would be very helpful.
