Why does Seifert-Van Kampen not hold with $n$-th homotopy groups? My question concerns the Seifert-Van Kampen theorem, in the following form.
Let $X$ be an arch-wise connected topological space, consider a poin $x_{0}\in X$, and let $\{U_{i}\}_{i\in I}$ be an open covering by archwise-connected open subsets of $X$ such that $x_{0}\in U_{i}$ for all $i\in I$, and $U_{i}\cap U_{j}\in \{U_{i}\}_{i\in I}$ for all $i,j\in I$. 
For any two elements of the covering $U_{i}$, $U_{j}$ such that $U_{i}\subseteq U_{j}$, let $\phi_{ij}:\pi_{1}(U_{i},x_{0})\longrightarrow \pi_{1}(U_{j},x_{0})$ be the map induced by the inclusion. Similarly, for any element of the covering $U_{i}$, let $\tau_{i}:\pi_{1}(U_{i},x_{0})\longrightarrow \pi_{1}(X,x_{0})$ be the map induced by the inclusion.
Let $H$ be a group and $\rho_{i}:\pi_{1}(U_{i},x_{0})\longrightarrow H$ any collection of homomorphisms defined for all $i\in I$ and such that if $U_{i}\subseteq U_{j}$ then $\rho_{i}=\rho_{j}\circ \phi_{ij}$. Then, there exists a unique homomorphism $\lambda:\pi_{1}(X,x_{0})\longrightarrow H$ such that $\rho_{i} = \lambda \circ \tau_{i}$, for all $i\in I$. Moreover, this universal mapping condition characterizes $\pi_{1}(X,x_{0})$ up to a unique isomorphism.
It is well-known that this statement does not always hold if we consider $n$-th homotopy groups with $n\geq 2$, and I believe that easy examples of this fact could be constructed. However, I would like to understand the intrinsic reason behind this phenomenon. We can roughly say that the proof of Seifert-Van Kampen is divided in two parts.
(1) With the notation above, the group $\pi_{1}(X,x_{0})$ is generated by the union of the images of $\pi_{1}(U_{i},x_{0})$ via the maps $\tau_{i}$.
(2) For any $[\alpha]\in\pi_{1}(X,x_{0})$, consider the decomposition $[\alpha]=  [\alpha_{1}]...[\alpha_{n}]$ given in point (1). Then the morphism $\lambda([\alpha])=\rho_{1}([\alpha_{1}])...\rho_{n}([\alpha_{n}])$ is well-defined.
Which one of the two parts fails with $n$-th homotopy groups? Why?
 A: You ask a very subtle question! 
One of the intrinsic reasons is that in homotopy theory identifications in low dimensions in a space usually have high dimensional  influences. An easy   example is to attach a whisker to a sphere, i.e. form $X=S^n \vee [0.1], n >1$. This has the same homotopy type as $S^n$. Now identify $0$ and $1$ in $X$, an identification in dimension $0$, to form $S^n \vee S^1$. The $n$-th homotopy group has changed! There are many more complicated examples, and this is the norm. 
The answer which I have pursued over decades is that one needs "invariants"  which have structure in a range of dimensions. How to find such, and prove theorems?  It turns out that forms of multiple groupoids do a useful job, but they are nicely defined, and lead to calculations,  not for spaces but for spaces with structure.  The structures that have worked in this way are filtered spaces and $n$-cubes of spaces. See my presentation Galway as a start, but that explains only filtered spaces. 
Part of the confusion is that a space with base point seems much like a space, but maybe it should be seen as profoundly different! Homotopy groups are defined for the former but not for the latter. So one is looking for functors 
$\Pi: ($topological data$) \to ($algebraic data$) $ 
which preserve certain colimits. It turns out that the latter algebraic data come in many equivalent forms when including dimensions $>1$, partly due to the many forms of convex sets in Euclidean space of dimensions $>1$. 
With regard to your questions $1,2$, in higher dimensions you have to consider higher dimensional compositions, and these are best expressed cubically. Thus in dimension $2$ we consider the following diagrams.

Going from left to right is subdivision, but going from right to left should be composition. We need algebraic inverses to subdivision and a homotopically defined gadget which can express this. How we got to one successful answer to this in all dimensions is described in the presentation Galway. Note that "double groups" are just abelian groups, by the interchange law (i.e. Eckmann-Hilton argument),  while double groupoids are much more complicated than groups, and so better able to express the complications of homotopy theory in dimension $2$. The book Nonabelian Algebraic Topology (pdf available) explains history and intuitions, before developing the theory.
It also turns out that one needs two kinds of algebraic data, which I have termed "broad" and "narrow" in the Galway talk. The broad data is for conjecturing and proving theorems, while the narrow data is used for relation to the classical theory, and for calculations. A substantial part of the work is to prove algebraically that these two categories of data are equivalent, so that one can hop from one to the other as desired. 
Geometrically, these two kinds of data are related to the use of cubes and disks in defining homotopical constructions, such as homotopy groups. All this works nicely if the topological data are filtered spaces, where $\Pi X_*$, with $X_*$ a filtered space, is constructed using fundamental groupoids and relative homotopy groups, giving crossed complexes, the narrow model, while a less familiar construction, $\rho X_*$, modelled on cubes, gives the "broad" model, which allows us to express composition and subdivision in many dimensions. Cubes are also very helpful in discussing products and homotopies, because of the rule $I^m \times I^n \cong I^{m+n}$. 
The above enables the  homotopical approach to basic algebraic topology, without using singular homology, described in the  book Nonabelian Algebraic Topology. To get further one needs some much more complicated gadgets, called cat$^n$-groups, and crossed $n$-cubes of groups, for which there is a survey article here. There have been complaints about this theory, for example that it was  unsuccessful at calculating higher homotopy groups of spheres, but few topologists have worked on it;   it enables calculations not previously available of homotopy types of some complexes, while group theorists have liked the nonabelian tensor product which arose from some pushout calculations, is related to commutator theory,   and now has a bibliography of 174 items. 
Nov 16, 2019 The first mention of higher homotopy groups seems to hasve been at the 1932 ICM at Zurich, in a seminar of E.Cech. He proved they were abelian for $n\geq 2$, and on these grounds it was argued that they were not the hoped for at that time higher dimensional versions of the fundamental group. For accounts of this see books on the history of topology. Now we know that higher homotopy groupoids can be seen as a more reasonable such generalisation.  See the above references and also this, and another. 
