As for point (1), maybe the following references will be useful:
https://www.researchgate.net/publication/228707872_The_Euler-Maclaurin_formula_revisited (The Euler-Maclaurin formula revisited, by D. Elliott)
http://link.springer.com/article/10.1007%2Fs002110050392 (The Euler-Maclaurin expansion and finite-part integrals, by G. Monegato, J.N. Lyness) and references in it, especially [8] (I. Navot) and [9] (B.W. Ninham).
The Euler–Maclaurin formula revisited, by D. Elliott
The Euler–Maclaurin expansion and finite-part integrals, by G. Monegato, J.N. Lyness and references in it, especially [8] (I. Navot) and [9] (B.W. Ninham).
