Part of my question has been asked before (Every non-negative multivariate polynomial has degree even?) but the proof there is not very satisfactory. The other part of my question involves proving (or disproving) that the highest-degree term, say $ c\prod_{i=1}^n x_i^{a_i}$ (where we know $\sum_i a_i$ is even), must have its coefficient $c > 0$.
Are these two conditions (i.e., the polynomial having even degree and its highest degree term having positive coefficient) also sufficient for guaranteeing the polynomial is bounded from below?
Let $R(x_1, \ldots, x_n)$ be our polynomial, $P(x_1, \ldots, x_n)$ be a polynomial consisting of highest degree monomials from $R$ (it's homogeneous polynomial of odd degree) and $Q = R - P$.
We need that any polynomial with non-zero coefficients is non zero in at least one point. We can prove it by induction by number of variables: non-zero polynomials of $1$ variable have finite number of zeros; if polynomial of $n + 1$ variables is zero everywhere, then coefficients of any degree of the first variable are identically zero as polynomials of rest $n$ variables.
So for some $a_i$, $P(a_1, a_2, \ldots, a_n) \neq 0$. Thus $P(a_1 t, a_2 t, \ldots, a_n t)$ is non-zero homogeneous polynomial of one variable of odd degree. Then $R(a_1 t, \ldots, a_n t)$ is polynomial of odd degree - then it is negative at some point $t_0$, and so $R$ is negative at $a_1 t_0, a_2 t_0, \ldots, a_n t_0$.
For the second part - if highest degree term is unique and polynomial is positive - highest degree coefficient should be positive (otherwise polynomial will approach $-\infty$ as variables grow). It it's non unique - some coefficients can be negative. Take, for example, polynomial $(x - y)^2 + 1$.
Having even degree and all highest degree coefficients positive isn't enough for polynomial to be bounded: take, for example, $x^2 - y$.