Counting Lattice Points with Ehrhart Polynomials

For question 3, by the reciprocity law $(-1)^{n}P(-t)$ counts positive integer solutions of $\sum x_i/b_i<t$ where $P$ is the Ehrhart plynomial of the $n$-simplex with vertices $(0,\ldots,0)$ and the $(0,\ldots,b_i,\ldots,0)$. This is short of your target by the number of positive integer solutions of $\sum x_i/b_i=t$. But that is $(-1)^{n-1}Q(-t)$ where $Q$ is the Ehrhart polynomial of the $(n-1)$-simplex with vertices the $(0,\ldots,b_i,\ldots,0)$.

(Partial Answer to Question 1) No, this phenomenon isn't restricted to integral convex polytopes. There are examples of non-integral rational convex polytopes with Ehrhart quasi-polynomials that are polynomials. See this reference.