I have seen the cohomological form of the index theorem usually stated in the following form: $$ \int_X \varphi^{-1}\left(\operatorname{ch}([\sigma(P)])\right).\operatorname{todd}(TX\otimes\mathbb C) $$ where $\varphi$ is the Thom isomorphism, or as $$ \int_{T^*X}\operatorname{ch}([\sigma(P)]).\operatorname{todd}(TX\otimes\mathbb C). $$ My question is: when are both formulations equivalent and how to show that they are? (For example, $X$ clearly needs to be orientable in order for the first formulation to make sense ...)
A related point that also bothers me is that the second formulation involves a Todd class of an element of $K^0(X)$ and not $K^0(T^*X).$ Is this to be interpreted as seeing $K^0(T^*X)$ as a right $K^0(X)$-module with the module structure arising by pulling back along the projection map $T^*X\to X?$
And then there is a final little thing that bothers me: sometimes, eg. in the the book "Topology and Analysis: the Atiyah-Singer Index Formula and Gauge-Theoretical Physics" a factor of $(-1)^n$ or similar appears on the RHS, while otherplaces, eg. the Wikipedia page for the index theorem, the correction coefficient is absent. Which is the correct version?
Thanks!