The space of polynomials with all real roots

I must be missing something. A monic polynomial with real roots has nonnegative coefficients if and only if all roots are nonpositive. If we have two vectors $u,v\in \mathbb{R}^n$ with nonpositive entries, then there is a continuous path (e.g., a line segment) from $u$ to $v$ that keeps the entries nonnegative. This induces a continuous path between the polynomials with roots $u$ and $v$ that keeps the roots nonpositive.

Space $R$ of hyperbolic(i.e. real-rooted) polynomials of one variable, have actually been studied quite a lot, mostly in papers by V.I. Arnold and his students at the end of 80's-beginning of 90's

I would recommend a book by V.P. Kostov Topics in hyperbolic polynomials of one variable. Societe Mathematique de France, 2011, especially it's chapter 2.

It has a lot of references.

Moreover, Section 2. of a recent preprint http://arxiv.org/abs/1512.08645 has some review on this and some other close questions.

In particular, one can prove that $R$ is homeomorphic to ${\mathbb R}\times{\mathbb R_+^{n-1}}.$

This follows from the fact that the correspondence between roots and coefficients of polynomials is morphism of symmetric product of topological spaces (i.e. quotient by $S_n$ acting by permutations of roots -- see http://www.ams.org/journals/tran/1954-077-03/S0002-9947-1954-0065924-2/S0002-9947-1954-0065924-2.pdf P.538 or http://link.springer.com/article/10.1007%2FBF01094483) and $n$-th symmetric product of real line is exactly ${\mathbb R}\times{\mathbb R_+^{n-1}}$ (see e.g. Proposition 3 there: https://ncatlab.org/nlab/show/symmetric+product+of+circles).