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Historically first reference (as I know) positively answering on your question 2 1 is the appendix by F. Laudenbach in the paper:

Bismut, Jean-Michel; Zhang, Weiping An extension of a theorem by Cheeger and Müller. (French summary) With an appendix by François Laudenbach. Astérisque No. 205 (1992),

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Historically first reference (as I know) positively answering on your question 2 is the appendix by F. Laudenbach in the paper:

Bismut, Jean-Michel; Zhang, Weiping An extension of a theorem by Cheeger and Müller. (French summary) With an appendix by François Laudenbach. Astérisque No. 205 (1992),