By $X\xrightarrow{f}Z\xleftarrow{g}Y$, we can define the homotopy pullback $X\times_Z^hY$.
Now if we have $X'\xrightarrow{H}X\xrightarrow{f}Z\xleftarrow{g}Y$, where $H:X'\rightarrow X$ is a homotopy equivalence, we can define another homotopy pullback $X'\times_Z^hY$. Then are $X\times_Z^hY$ and $X'\times_Z^hY$ also homotopy equivalent?
More generally, if the homotopy limits(colimits) respect homotopy equivalences?
To answer your first question, yes.
There are a pair of naturally induced comparison maps $X'\times_Z^hY\rightarrow X'\times_X^h(X\times_Z^hY)$ and $X'\times_X^h(X\times_Z^hY)\rightarrow X\times_Z^hY$, which in this case are both homotopy equivalences.
That the first is so follows because the basic rules of pullback/pushout squares carry over to homotopy pullback/pushout squares. In particular you have a pasting rule stating that given two adjacent squares $[A]$, $[B]$ sharing a common edge, then if $[A]$, $[B]$ are homotopy pullback/pushouts then the overall square $[AB]$ is a homotopy pullback/pushout. I apologise if that sounds a little vague, but I'm struggling to draw a diagram. A better statement can be found in Proposition 3.3 here, https://ncatlab.org/nlab/show/pullback.
The second comparison map is a homotopy equivalence because it is the homotopy pullback of a homotopy equivalence. Whilst it is not true in a general model category that the pullback of a homotopy equivalence will be a homotopy equivalence, it is true that the pullback by a homotopy equivalence of a fibration between fibrant objects will also be a homotopy equivalence. Since the square is a homotopy pullback, these conditions certainly hold sufficiently for the map $X\times_Z^hY\rightarrow X$ to be able to get the claim.
For your second question, also yes. The point of introducing homotopy limits/colimits is to obtain a homotopy invariant replacement for the notion of limit/colimit. If $\mathcal{I}\xrightarrow{F,G}\mathcal{M}$ are two weakly equivalent diagrams in a model category $\mathcal{M}$ then $\text{holim}\; F\simeq \text{holim}\; G$ and $\text{hocolim}\; F\simeq \text{hocolim}\; G$. Generally, but not always, weakly equivalence in functor categories is defined pointwise. That is, a natural transformation $\eta:F\Rightarrow G$ is a pointwise weak equivalence if and only if each of its components $\eta_i:F(i)\xrightarrow{\simeq}G(i)$ is a weak equivalence in $\mathcal{M}$. I might suggest reading Hirschhorn's book "Model Categories and Their Localizations" for an introduction to the required machinery. A more direct introduction might come from Dugger's notes "A Primer On Homotopy Colimits".