Albeit rather late I would like to add the following reference as another answer to your question (1): dimensional regularization was 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).

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).

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

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

Albeit rather late I would like to add the following reference as another answer to your question (1): dimensional regularization was 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).