Question 1: One of the easiest ways to see this 3-fold intersection condition is in terms of "brick subdivisions" of the square. This is a subdivision of the square into rectangles such that each vertex of the subdivision is a corner of 3-rectangles. Such a subdivision can be taken fine enough to carry out the argument given in Hatcher's book. More details are in
R. Brown and A. Razak Salleh, ``A van Kampen theorem for unions of non-connected spaces'', Archiv. Math. 42 (1984) 85-88.
which relates this argument to the Lebesgue covering dimension. Also by using the fundamental groupoid $\pi_1(X,A)$ on a set $A$ of base points, it gives a theorem for the union of non-connected spaces.
I also prefer the argument in terms of "verifying the universal property" rather than looking at relations defining the kernel of a morphism. To see this proof in the case of the union of 2 sets, see
One of the problems in the proof for the fundamental group as given by May is that it does not generalise to higher dimensions.
Question 2: One has to ask: where do such unions of non-connected spaces arise? The standard answer, and for me from which all this groupoid work arose, is to obtain the fundamental group of the circle $S^1$, which is after all, THE basic example in algebraic topology. Many other examples arise in geometric topology.
More answers come in applications to group theory; for example the Kurosh subgroup theorem on subgroups of free products of groups can be proved by using a covering space of a one-point union of spaces. The cover over each space of this union has usually many components. I confess I have not seen the minimal condition of 3-fold intersection used in practice, but it is always interesting to know the minimal conditions for a theorem. Just as well to know one can't in general get away with 2-fold intersections.
You can see the consistent use of groupoids in $1$-dimensional homotopy theory (van Kampen theorem, homotopy theory, covering spaces, orbit spaces) in my book Topology and Groupoids (2006) available from amazon.
Edit Feb 17, 2014:
It may be useful to point out that a sharper result than that in May's book was published in 1984 in the above Brown-Razak Salleh paper; the idea of proving a pushout result for the full fundamental groupoid, and then retracting to $\pi_1(X,A)$, is in my 1967 paper, and subsequent book, but Peter May cleverly makes it work for infinite covers; the problem is that this proof does not generalise to higher dimensions, as far as I can see.
Also Munkres' book on "Topology" uses non path connected spaces in dealing with the Jordan Curve Theorem, and for this uses covering space rather than groupoid arguments. I do not see why books avoid $\pi_1(X,A)$, since it hardly requires any extra in the proofs to that for $\pi_1(X,a)$, given the easy definition.
Edit Feb 20, 2015: A relevant question and answer is http://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one
Some of the relevance to the history of algebraic topology, and to further developments, is in this presentation in Galway, December, 2014.
March 5, 2015
I feel this "perturbation (or deformation) trick" is of quite a fundamental nature. Above is a picture of part of a deformation involved in the proof of the 2-dimensional Seifert-van Kampen type theorem, for the crossed module involving a triple $(X,A,C)$ of spaces where $C$ is a set of base points, and $C \subseteq A \subseteq X$.
The red dots denote points of $C$.
The blue lines denote paths lying in $A$. The small squares in the bottom are supposed to denote squares lying in a set $U$ of an open cover. By making connectivity assumptions you can deform a small bottom square into the top one of the right kind, and still lying in the same open set. So, by working with a double groupoid construction $\rho_2(X,A,C)$ whose compositions are more 2-directional than the usual relative homotopy groups, you can get going with the proof of a universal property involving double groupoids, and hence for crossed modules.