As for “cook up well”, what kind of apple cooks up well depends entirely on what you are cooking. The category of “cooking apples” is best broken down into “sauce apples” and “pie apples.”
If you are making an apple sauce or an apple butter, you will want an apple that breaks down easily into a purée.
If you are making apples pies, a compote, or if you are stewing apples, or baking them whole, you will want an apple that can hold its shape when cooked. You don’t want to base your apple pie if you can help it on an apple that will break down into a purée, because then the pie will appear flat after cooking, and when a slice is lifted away from the pie dish, the filling (which has turned to sauce) can drip out in mid-air.
Cooking apples generally tend to also be tart apples, as cooking gives you a chance to balance their pucker power with a sweetener to provide a range of flavour.
There’s another, more banal consideration, too. What variety of apples constitutes a good cooking apple will also depend on what you can realistically buy where you are. Wealthy Apples may be a preferred pie apple amongst old-timers in the eastern Canada and US, but in Australia, cooks will have a much easier time getting their hands on Granny Smiths for their pies.
Apple sauce; canned apples; another fruit.