That is, if a monster cohort levels up with a PC and they would normally qualify for a class ability, like the fighter's bravery or a barbarian's rage, do the monster cohorts also get these abilities?
Do also get feats and skills just like a leveling character would or is the class leveling of monster cohorts mostly just for the sake of advancing their usual stats?
When a Creature Gains Class Levels, It Gains the Class's Features
After all, that's why it takes class levels. But while the creature may possess the class features, some individual creatures may find using such class features difficult (e.g. creatures without the capacity for speech who take levels in a spellcasting class need to find alternative methods of fulfilling the verbal components; creatures without hands struggle to use many weapons despite a class granting it proficiency).
Monster Advancement's a Little Tricky
Creatures with more than 1 Hit Die from these Hit Dice gain feats and skills. When the creature gains 1 level in a class, it adds another 1 Hit Die. A creature's Hit Dice equals its character level for determining its feats and maximum skill ranks. You can use this table to help you determine, for instance, how many feats a creature with class levels has by adding the class levels to the creature's Hit Dice. For example, a hill giant has 10 Hit Dice. Were it to take 1 level of oracle, it would then have 11 HD and be eligible for another feat.
