Kohlberg & Mertens (Econometrica 1986) Show that the graph of the Nash Equilibrium correspondence is homotopic to a homeomorphism (Theorem 1). They also give an example of an particular game where the set of NE's is homeomorphic to a circle.

$x_i=1$ is the only feasible point unless you assume the rows of $P$ are linearly dependent.

There is a literature on these kind of issues started by Hoeffding (1963). On page 360 of Marshal and Olkin "Theory of Inequalities and Majorization" you can see some results that may be of help (due to Karlin and Novikoff).

Sorry too many square roots in the last inequality - it should read $\sum_i\sqrt{x_i}\geq \prod_ix_i^{-2x_i}$!

If you use $\log\sum_i \sqrt{x_i} = \log\sum_i x_i \frac{1}{\sqrt{x_i}}\geq \sum_i x_i \log(\frac{1}{\sqrt{x_i}})=-2\sum_ix_i\log x_i$ (where the inequality is Jensen's) you get $\sum_i \sqrt{x_i}\geq \sqrt{\prod_i x_i^{-2x_i}} $. Which is enough for the lower bound, because $\prod_i x_i^{x_i}<1$.