Well, a cockroach is likely to be almost identical to every other cockroach out there. Even if it wasn't, it probably wouldn't matter that much.
Domestic animals might only have slight personality differentiation. Some will be nicer than others, some braver, some happier, etc. Not much variation, but a little.
A dog on the other hand--well, we've all had or known dogs with a lot of personality, and some more intelligent than others. Cats too--Scamps for example. So they might display a wide range of emotions and characteristics, they'd just be lacking in human-level intelligence and sophistication.
The next step up would be the Slow Learners. They'd be able to use and make tools, and may even have a crude language (or be able to learn one), but would lack in more abstract thought. Apes and trolls would fall under this category. They'd have distinct personalities, which would be more sophisticated and complex than our pets.
More than human intelligence might indicate a being that was able to instantly learn skills and devices to which it was introduced, and to remember everything eidetically, to get automatic bonuses to anything it attempted-including melee combat-to consistently create tools of masterwork, or even Artifact quality, and to predict with some accuracy, the likely outcome of a series of seemingly random events. Their personalities might be unpredictible, or even multiple.