There are tons of different cat/dog repellants on the market - check any of the usual online pet supply places, like Petsmart, or http://www.revivalanimal.com . Some of them smell worse than cat urine, though! You might have to try several different products to find what works, as some cats hate, for example, citrus, and some don’t mind it at all.
If you can afford it, a ScatMat (for indoors) is great. This is a pad that gives a tiny electric shock when touched (nothing drastic or harmful) and they are sold in many different sizes. Usually you can use one for a while to re-train, then put it on a shelf until needed again. I think they originated as Christmas tree protection.
You might try Oxyclean to remove odors from your carpet or upholstery - I’ve started using it recently and it seems to work well. Just make sure you test an area first, and don’t just dump a bunch in some water on the theory that more is better - it ate holes in some blue jeans I was trying to un-stain!
Deacons, could your cat be trying to dig under the cellar door because she is hearing/smelling a mouse (or some other interesting creature)? I wouldn’t let her through the door to find out, as she may think she has learned how to ask you to open doors! I’m not sure what you can do about this (other than pest control), or aversion training - spray gun, loud noise, ScatMat, etc. But you might try tacking a piece of that vinyl carpet protector down over the carpet by the door, allowing it to extend under the door a little ways so she can’t catch the edge and pull it loose.
Cats will urinate outside of the litter box for a number of reasons, and it may require some detective work to find out why. And once the behavior starts, for whatever reason, it can become a habit if it goes on for some time.
First thing, always, is a vet check, especially if a cat has previously been very reliable. A bladder or kidney infection can cause painful urination, which the cat might associate with the litter box and thus seek relief in other places. An infection might also make it difficult for the cat to hold its urine long enough to reach the litter box.
Second is to evaluate how clean you are keeping the litter box - some cats are really picky about using what they consider to be a dirty litter box. Maybe you used to be at home a lot and scooped twice a day, but now you are working so only have time for once? Nothing wrong with that, but your cat may be used to a cleaner box and is unhappy with the ‘new schedule’.
Next, location - have you moved the litter box from a previous location, even if only a foot or two? Some cats are very particular about that. Have the food and water dishes been moved closer to the litter box? That, also, will put some cats off. Has something else in the vicinity of the litter box changed - moved furniture, for example?
Fourth on the list - type or lack of litter and/or box. Have you changed the brand or scent of litter you use? Gone from clay to clumping or something else? Or has the litter you’ve been using ‘improved’, and maybe your cat doesn’t like it? Some cats will use anything they can scratch in, while others can be incredibly picky, and even a new scent added to the old litter can put them off. And some cats prefer no litter at all - they like to pee on solid, slick surfaces (hence their tendency to use bathtubs/showers). Changing litter boxes can also put a cat off - some cats will not use a covered litter box, some will not get within 10 feet of a LitterMaid, etc. If you’ve changed the litter box in any way, try going back to the old style and see if the situation improves. Oh, and number of litter boxes per cat - some cats don’t like to share, the ideal number is one box per cat, plus one extra, all at different locations.
Then we have behavioral issues of several varieties. Most common is territorial marking, often brought on by some change in the household that makes the cat feel insecure. A new pet, a baby, a visiting relative, a stray cat that is prowling around you house, new carpet or furniture, new, loud neighbors - any change can cause a naturally nervous cat (often inherited, so don’t feel guilty) to feel insecure. BTW, cats that are allowed to go outside are worse about territorial marking than inside-only cats. In-and-out cats that are forced to become inside-only often exhibit worse marking behavior for a couple of weeks, then taper off until it is no longer a problem.
Could the cat have been frightened by something while in the litter box, and is now reluctant to use it? Especially a covered litterbox, where a cat may feel trapped. Young children or another pet may be bothering the cat while it is in the box. A sudden loud noise or something dropped on/in the box at the wrong time, etc. - anything that could have caused pain or fear while the cat was in the box might make it reluctant to risk that again.
Suggestions: A vet check, in case this is a health-related problem. Even if it is a health problem, inappropriate elimination may have become a habit (reinforced by lingering odors) and will require some retraining. I suggest confining the cat to a single room with several litter boxes, maybe with different types of litter and/or boxes,if you suspect that might be the problem. Be observant - does the cat seem to prefer one litter box, or one litter? Does the cat use each box once (indicating a preference for an extremely clean litter box)? Does the cat continue to urinate outside of the litter box, or will it use a litter box reliably while confined? All of this will provide clues for your next best option.
If/when the cat is using the litter box well, expand the area of confinement gradually, if possible, before allowing free run of your household. If the cat reverts to the old behavior, step back to the previous confinement level until he/she is reliable once again. If your cat has a preference for one single spot, place a litter box on top of that spot, if possible.
Occasionally we breeders have a kitten that just doesn’t seem willing to accept a litter box, and we’ve successfully trained them by placing them in a large cage and covering the entire floor with litter. They have no choice but to use the litter, it becomes a habit in a very short time, and afterwards they will seek out a litter box with no problem.
Your vet might also prescribe antidepressant or tranquilizing medication during the retraining period, which often helps with stubborn cases. (Remember that you have to give the cat a pill every day before asking for this!)
BTW, I highly recommend the new ‘crystal’ litters, if your cat doesn’t mind using it (switch over gradually!). It absorbs urine (and its odor) immediately and completely, reduces the odor and mess from feces, and lasts much longer than any other type.