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I'm sorry she's not always urinating in the litterbox. I know how frustrating this can be. Even though you don't believe this is a urinary tract problem, in cases such as these, a in-person exam, urinalysis and urine culture, is always recommended first, to rule out any underlying medical problems.
With 4 cats, you should have 4 or 5 boxes, if you have the room, as the general rule is 1 cat per box, plus one. None of them should be covered and use an unscented clumping litter. You can try 'Cat Attract' litter or 'Cat Attract' litter additive, which you add to her regular litter, as it contains herbal attractants that encourage use of the box. Here is more info:
Many times, the cause for eliminating outside the litterbox is stress or anxiety, so get some Feliway plug in diffusers, which emit cat calming pheromones and are effective in helping with inappoprirate elimination.
Because you have a 4 story house, in addition to the boxes in the basement, there should be a box on every level of the house.
Make sure to clean all areas outside the box that she has soiled, with an enzyme based product like Nature's Miracle, Zero Odor, Odor Mute, or many others that are available in y our pet supply store. If the areas are not cleaned well enough to completely remove the odor, she will return there, thinking it is an okay toileting place.
The fact that she goes in the tub and on your bed (Nature's Miracle makes a laundry detergent and laundry 'booster' additive for washables) may mean she is experiencing pain, because she is seeking out a soft surface and a hard, cool surface, thinking if she feels that under her paws it won't hurt to go; she could be associating the box itself with discomfort/pain when she uses it.
Cats who are used to going outside sometimes have difficult adapting to the litterboxes inside. Some cats like the feel of sandy soil to scratch in, when they go. You could try to fill a couple of boxes with sand, sold in major children's toy stores, as used for sandboxes. She might prefer this substrate.
I usually prefer to advise the use of a prescription medication as a last resort, but discuss with your vet, Clomicalm and other medications that help with stress/anxiety and inappropriate elimination. Here is information about Clomicalm:
I hope things improve with Luna and remember to have her checked out by the vet, first.
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