Regarding question 1: An example from combinatorics (or conceivably topology, depending on how one classifies these things): if a finite simplicial complex is Cohen-Macaulay (defined below), then its $h$-polynomial (also defined below) has nonnegative coefficients. So this is one example of a situation where you can detect that an object of some sort lacks some nice property by observing that an associated polynomial does not have nonnegative coefficients.
Terminology review, as promised above: given a finite $(d-1)$-dimensional simplicial complex $\Delta $, let $f_i(\Delta )$ be the number of $i$-dimensional faces in $\Delta $, making the convention that the empty set is a $(-1)$-dimensional face. The vector $(f_{-1}(\Delta ),f_0(\Delta ),\dots )$ is called the $f$-vector of $\Delta $. One may encode the same data with another vector called the $h$-vector of $\Delta $ given by $h_k(\Delta ) = \sum_{i=0}^k (-1)^{k-i}{d-i\choose k-i}f_{i-1}(\Delta )$ for each $0\le k\le d$, so for instance $h_d(\Delta )$ is the reduced Euler characteristic of $\Delta $. The $h$-polynomial of $\Delta $ is the polynomial $h(x) = h_0 x^d + h_1 x^{d-1} + \cdots + h_d $, while the $f$-polynomial of $\Delta $ is $f(x) = f_{-1}x^d + f_0x^{d-1} + \cdots + f_{d-1}$. Now an equivalent way (to above) of describing the relationship between $h$-vectors and $f$-vectors is via $\sum_{i=0}^d f_{i-1}(t-1)^{d-i} = \sum_{i=0}^d h_it^{d-i}$, or in other words as $h(x)=f(x-1)$.
Now a finite $d$-dimensional simplicial complex $\Delta $ is said to be Cohen-Macaulay over the integers if all its reduced homology groups satisfy $\tilde{H_i}(\Delta ,\mathbf{Z})=0$ for $i < d$ and likewise the link $lk_{\Delta }(\sigma )$ of any face $\sigma \in \Delta $ has reduced homology groups satisfying $\tilde{H_i}(lk_{\Delta }(\sigma )) = 0$ for $i < dim (lk_{\Delta }(\sigma ))$. Cohen-Macaulay complexes have the property that their $h$-vector is nonnegative, hence $h$-polynomial has nonnegative coefficients. A good reference for this and related topological combinatorics is the book "Combinatorics and commutative algebra" by Richard Stanley.
Regarding question 2: An important way of constructing positive polynomials is as sums of squares. Many examples of positive polynomials with nonnegative coefficients may be constructed this way -- if the smaller polynomials being squared have nonnegative coefficients too. However, not all positive polynomials are sums of squares by any means, let alone these special ones. A good reference on this is the paper:
G. Blekherman, Nonnegative polynomials and sums of squares, Jour. Amer. Math. Soc. 25 (2012), 617-635
which proves that there are many more nonnegative polynomials than sums of squares asymptotically by showing that the volumes of cross sections of the cone of nonnegative polynomials grow much faster as total degree is increased than the corresponding cross sections of the cone of sums of squares.
