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The first question appears to have an affirmative answer in Corollary 1 in Exotic Twisted Equivariant Cohomology of Loop Spaces, Twisted Bismut–Chern Character and T-Duality by Fei Han and Varghese Mathai. However this seems to conflict with the answer by Tom Goodwillie, and I am not yet able to exactly see what is causing this discrepancy. My guess is that actually two different versions of $LM$ as a topological space/infinite dimensional manifold occur in the two articles so that both statements are true relative to the version of $LM$ they assume. But this is just a quick guess.