Regarding 1: The Euler characteristic of a compact manifold M is the self-intersection of the diagonal ![$\Delta\subset M\times M$](http://latex.mathoverflow.net/png?%24%5CDelta%5Csubset%20M%5Ctimes%20M%24). If F is homotopic to the identity, then the diagonal (the graph of the identity) can be deformed to the graph of F. If F has no fixed points, this graph is disjoint from the diagonal; so the self-intersection, and thus the Euler characteristic, is 0.

Note that for any M, the product ![$M\times\mathbb{R}$](http://latex.mathoverflow.net/png?%24M%5Ctimes%5Cmathbb%7BR%7D%24) admits such an F, so the condition that M be compact is necessary.