Feynman diagrams are used in QED, quantum electrodynamics, they represent all possible interactions that may occur. A path integral is used to find the minimum energy for each interaction. The sum of these interaction energies is the total E. Without writing down the possible interactions of any particular situation the original theoretical eq. is useless, because the solution involves finding the energy that is a sum of all possible interactions. Without knowing the terms in the sum the actual sum can never be totaled. Each term in the sum involves finding the minimum energy for the particular interaction, that's where the path integral, or it's equivalent enters as a necessity.
Sorry but this is incorrect. Pick up a book on numerical methods and you'll find many ways of numerically solving differential equations that do not involve integrals or integrating.
All you've done is described in a round about way the perurbation expansion typically used in solving the QED equations. It is simply a mathematical method of solving the original equations.
I should also point out that even if a numerical approach to the QED equations were to yield a set of integrals, these would tyically *NOT* be the same set of integrals that correspond to Feynman diagrams.
Eulers method is a simple method of numerically solving differential equations that does not involve an integral in any sense.