Next, a longer survery article from the 2006 ICM, aptly titled Tropical Geometry and its Applications, where he covers your three mentioned applications, extends the idea of curves in the previous article to general varieties, and then gives nice ways to compute Gromov-Witten invariants of $\mathbb{CP}^2$. If you like real algebraic geometry, he also computes Welshinger numbers, and methods of gluing hypersurfaces of toric varieties in higher dimensions.