Ball joints often make your steering feel either too tight or too loose, depending on how the suspension is wearing. Look at the rubber boots to see if any grease is oozing out, or if the boot is cracked or even missing. You might feel "tight spots" in the steering, or hear an obnoxious creaking sound when you turn the wheel if they're really getting bad. As for replacing them, depending on your generation, they might be either pressed in or bolted on (I'm not too familiar with the setup of late-model Corollas). If you're driving a 7th-gen car ('93-97), then I can tell you exactly how to do it, as they're bolted on. I just finished up replacing both the CV axles and ball joints on my own car.
As for KYB struts, I have not personally had any experience with them, but I do know they came on some Corollas as OEM equipment from the factory. If you're looking to return the ride and handling to factory specs, then I'm sure they would be a good choice. Just don't slap on some lowering springs with them, as they won't be matched to the shorter springs.