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, ``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 on a set 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 then looking at relations defining the kernel of a morphism. To see this proof in the case of the union of 2 sets, see
Question 2: One has to ask: where do such unions of non-connected spaces arise? One answer
is 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, 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 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 to $\pi_1(X,a)$ in the proofs, given the easy definition.