First passage percolation is a simple model describing a random distance on a graph, classically the square lattice, obtained by assigning each edge an i.i.d. weight. Under weak assumptions on the edge weights, the distance converges to a deterministic large-scale limit and balls with respect to this distance converge to a deterministic limit shape. Due to progress made for related models in the so-called KPZ universality class, there is a fascinating discrepancy between the set of conjectures for FPP and the set of rigorously proven results. In this thesis, we slightly expand on the latter by providing an improved lower bound of log(|x|)^1/2 for the non-random fluctuations, i.e., the difference between the expected random distance between the origin and a vertex x and its large-scale limit. The bound holds for edge weight distributions with small support, but could be extended if we can prove that the limit shape is not a polygon. We show that this condition can be verified for a simplified version of FPP, which brings us to the second part of this thesis. Let the vertical edge weights now be deterministic. Then the limit shape has a flat edge in the vertical direction if and only if the weight distribution on the horizontal edges has an atom at the infimum of its support. We also investigate another interesting property of this simplified model known as shear invariance and hope that our discoveries could bring further insight into the standard model as well.