Should we start from Euclidean QFT if we are to be rigorous? Path integral is only rigorous in Euclidean QFT. This suggests that one should start from Eucliden QFT and transport back the results back into Minkowski time. Is this how I should think of QFT?
 A: There are many many ways to build QFT from, more or less complete and rigorous. It is true that the most common rigorous axiomatization of QFT in the path integral formalism, the Osterwalder-Schrader axioms, are done in Euclidian space, as well as Reed-Simon's construction of QFT. 
This isn't to say that there are no methods to do it in Minkowski space, or more generally Lorentzian manifolds (which is important as you can't Wick rotate a general spacetime). There have been various attempt to define path integrals in the complex regime, such as Cécile Dewitt's integrators[1][2].
A: Personally, I found how Feynman explained the construction of the Path Integral the most intuitive as it went directly from physical principles.
Although it's important to make things rigorous it might be worth recalling here that mathematics still hasn't made the ordinary variational calculus that Bernouilli and Newton used over three hundred years fully rigorous. 
If I recall correctly, the only rigorously corrected QFT is the free QFT. That is with no interactions. 
