Gravity with Character physics is controlled by the Fall Speed Max setting in Physics properties panel, and you need to set it very low, in order to keep gravity from cancelling out your upward momentum. You can also use large z-motion values, but at default settings, gravity will still overtake upward momentum, resulting in a jump (convenient, that).
As for using other physics types, if you want more realistic physics they might be a good choice, but character physics may still suit your needs better. While they are incredibly useful (almost obligatory) in making platformers, they are also have benefits for other game types: By deviating from realistic physics, they can often be easier for players to control, particularly when it comes to collision behaviors. Rigid Body objects tend to flip over during collisions unless you lock rotation, in which case they behave like Dynamic objects and bounce off of walls. Character physics tend to react in more traditional ways.