Does gaining a level in a prestige class in Pathfinder (1e) give you feats and ability score improvements? Or do you only gain those from "base" classes?
The base feats gained at odd levels, and the ability bumps gained every four levels, are a function of hit dice of any kind. Creatures get them as they advance and gain hit dice, and characters get them as they take class levels, since a hit die is one of the benefits of every class level.
The type of hit dice does not matter for these purposes, just the number of them. Multiclassing, whether with base classes or prestige classes, does not affect things either—when you get enough HD, you get these basic benefits.
Pathfinder prestige classes are, with few exceptions, rather weak. They would be basically unusable if they also blocked your basic feat acquisition.