Question about the foundation of part I in A. Zee's book Zee says in Section I.3 of QFT in a nutshell:

The functional integral
  $$Z = \int D \varphi e^{i \int d^4 x [\frac{1}{2} (\partial \varphi)^2 - V(\varphi) + J(x) \varphi (x)]}   \tag{11} $$
  is impossible to do except when 
  $$\mathcal L (\varphi) = \frac{1}{2} [(\partial \varphi)^2 - m^2 \varphi ^ 2].\tag{12}$$
The corresponding theory is called the free or Gaussian theory.

This restriction gives sudden birth to the Klein-Gordon equation and also practically allows the entire part I of the book to proceed as it is.  
So, two and a half questions:  


*

*How can I understand the necessity of this restriction? And, what does "impossible to do"  mean here? 

*As part I unfolds, Zee explains that the above is not so much "meant" to be solved, but rather, it is a generating functional. So, why is the restriction necessary in the first place?

 A: First of all, let me say the following: If anyone (perhaps you, @V.Moretti?) would be able to provide a more mathematically oriented perspective on this question, I think that would make a valuable complement to this answer, which can be characterized as pragmatic (or handwavey, depending who you'd ask!), rather than deep.
That being said, I will now answer both subquestions: 


*

*I think there is nothing all too deep behind this statement by Zee. In particular, I don't think he means to make a rigorous statement on the well-definedness of the path integral, and the way this may (or may not) depend on the particular form of $\mathcal L$. He simply means that the free field theory allows us to - sweeping all serious mathematical issues involved in defining the path integral measure etc. etc. under the rug - to explicitly perform this integration, as it all boils down to nothing more than (a generalized version of) the standard Gaussian integral $$ \int_\mathbb R e^{-x^2}\mathrm{d} x$$
In any serious model of the interactions between fundamental particles we have to (rather unsurprisingly) consider interaction terms. These spoil the - once again, ignoring many mathematical issues - simplicity of the integral, and are usually attacked by means of a perturbative expansion, as opposed to solving exactly.

*First considering a free field theory is instructive physically, because we usually expand around the free field theory (in the sense that we consider small/weak interactions). Furthermore, the mathematical treatment of interacting theories is quite similar, so the free field Lagrangian can be considered as an excellent warm-up exercise in this aspect as well, allowing the beginning student to build up some intuition and familiarity with common techniques.
