Hello again, and thanks for your additional and helpful information.
I'm glad to hear you've had routine lab tests done 3 months ago, because, due to her age, she'd be more prone to develop diabetes, hyperthyroidism, liver or renal issues, so bloodwork every 6 months is recommended. It's wise of you to plan on some GI tests, soon, because I feel this may be related to an intestinal problem or even an anal gland problem, more than a dislike of the brand of litter.
When a cat experiences discomfort upon elimination, she'll come to associate that pain with the place of elimination--the box--and choose to eliminate elsewhere. Usually cats in this situation, prefer hard, smooth surfaces, as they feel it will minimize or prevent the pain, so I'm not surprised she likes the smooth, hardwood floor.
A fecal analysis would be a good idea at this time, to see if there's any hidden blood in her stool, and also check for intestinal parasites.
If she's urinating in the box, she's not having a problem with the litter itself, but a good choice might be Cat Attract, if you haven't tried it yet. It has herbal attractants to encourage use of the litterbox. Also, if you haven't tried them yet, paper based Yesterday's News and S'wheat Scoop may feel softer on her paws, if she does have sensitive paw pads. A fine, sand-like scoopable litter, would be a better choice over clay and/or crystals, which may feel sharp under her feet.
Make sure to clean all areas she's soiled with an enzymatic cleaner to completely remove all scent:
Cats can also sometimes start eliminating inappropriately if there has been any kind of change in their lives or environment, as this causes stress and anxiety. If the reason for eliminating outside the litterbox is NOT found to be medical, it would have to be treated as behavioral.
Cats are creatures of habit and do not take kindly to change. If you have had any new people added to your household, or if a 'usual' person has left, if you got new furniture, carpet, appliances, another animal, etc., or if your usual daily hours at home and/or away from home have changed or you've gone on vacation recently, any of these situations could cause your kitty stress.
Feliway/Comfort Zone for Cats, is a cat-appeasing pheromone which can help to calm stressed cats and curb urine marking and/or inappropriate elimination. It comes in both a spray and plug-in diffuser. The diffuser plugs into an electrical outlet like an air freshener, but has no scent to humans. For more information, please see:
Major pet supply stores carry Feliway, you can also order it online, and some vet offices carry it.
The anal glands are located on either side of the anus, under the skin. If they become impacted or infected, this can cause discomfort, especially upon defecation, and this possibility should also be checked out by your vet. Sometimes, the anal glands just need to be emptied, to provide relief.
I'd suggest a second litterbox, also, as some cats prefer to urinate in one and defecate in another. Continue to keep both boxes clean at all times.
I hope you and your vet are able to get to the bottom of this problem soon, and your kitty will be fine!