Dimensional regularization in odd dimensions

There a number of papers by Alain Connes on Dimensional Regularization (Dim Reg) in the context of noncommutative field theory. Some of his papers cite

P. Breitenlohner and D. Maison, "Dimensional renormalization and the action principle," Comm. Math. Phys. Vol. 52, Number 1 (1977).

so I presume this might be a useful reference for you.

Regarding your observation, this is a well known fact among particle physicists and is regarded as a feature rather than a bug. Part of the standard lore is that Dim Reg replaces log divergences by poles but that linear and higher divergences often just give zero. Physicists are most interested in log divergences since these govern violations of scale invariance, determine beta functions and so on. In your example the integral has a pole at d=2 where the integral has a log divergence but vanishes at d=3 where the divergence is linear.

Albeit rather late I would like to add the following reference as another answer to your question (1): dimensional regularization is also treated rigorously in

P. Etingof, Note on Dimensional Regularization in Vol 1, pp 597–607, of Quantum Fields and Strings: A Course for Mathematicians (see also here).