Mayer-Vietoris Type Sequence For Pushouts
As an answer to your first two questions, the answer is yes if the pushout computes the so-called homotopy pushout.
In elementary terms this means that the map $C(f, g) \rightarrow P$ from the double mapping cylinder to the pushout is a homotopy equivalence. It is for example sufficient for one of $f, g$ to be a cofibration.
If this is the case, then we get a long exact sequence as Mayer Vietoris for the covering $U, V \subseteq C(f, g)$, where $U = X \cup A \times [0, \frac{2}{3})$, $V = Y \cup A \times (\frac{1}{3}, 1]$. Indeed, it is not hard to see that $U \simeq X, V \simeq A$ and $U \cap V \simeq A$.
If the pushout of your diagram is not a homotopy pushout, then I believe there is little hope for any long exact sequence. Suppose such thing would exist and would behave functorially with respect to map of diagrams. Then some form of five-lemma would imply that the obvious map from $D^{n} \leftarrow S^{n-1} \rightarrow D^{n}$ to $\{ * \} \leftarrow S^{n-1} \rightarrow \{ * \}$ must give a homology isomorphism on the pushouts, as level-wise this map of diagrams is a homotopy equivalence. But the pushout of the first diagram is $S^{n}$ and it is $\{ * \}$ for the second one, so their homology is different.
A nice exotic example is to take $A=X=Y=S^1$ and $f,g$ maps of degree $2$ and $3$ respectively, e.g. $z \mapsto z^n$ for $n=2,3$. Then the pushout $P$ is not even Hausdorff. hence we quickly replace the pushout by the double mapping cylinder, i.e. homotopy pushout, and get a nice CW-complex, whose fundamental group is is the trefoil group, $T$ with presentation $\{x,y\mid x^2=y^3\}$. A variation is to take the fundamental groupoid based on two points $0,1$ at each end of the mapping cylinder and get the fundamental groupoid on these two points having generators $x$ at $0$, $y$ at $1$ and $\iota :0 \to 1$, with the relation $\iota x^2= y^3 \iota$.
Nov 14,2020There is more on this in terms of exact sequences (not of homology) at (R, Brown, P.R. HEATH and H. KAMPS), ``Groupoids and the Mayer-Vietoris sequence'', — J. Pure Appl. Alg. 30 (1983) 109-129. This is useful for looking at pullbacks of covering maps.