Many cats develop litter box issues. In most cases, it's either a security or a medical issue. Deanna H. writes:
Deanna, if you haven't taken Loki to a veterinarian for a checkup yet, it might be a good time to make sure she's okay and has no problems with incontinence. Once she's gotten a clean bill of health, it's time to address her security issues.
Most cats have a period of adjustment after being adopted. In most cases, the worse their experiences with humans was before, the longer that adjustment period will be. Coming into a new home, especially one with a dog, can be very stressful. The more stress a cat feels, the more likely she is to have litter box issues.
The first thing to address is the litter box. If it's enclosed (has a lid) you want to remove the lid. Cats don't like feeling that they could be trapped where they're doing their business. They want to know there are multiple routes for escape. While your dog may be very friendly to Loki, she may still see him or her as a potential threat.
You also want to make sure that it's easy for Loki to get in and out of the litter box. Some cats have issues with high-sided boxes. You also want to avoid the new motorized contraptions that self-clean. The noise and unpredictable movement of the mechanism can be very scary.
The positioning of the box is also a big deal. You don't want to put it in a noisy or high-traffic area but it needs to be easily accessible. If you put it in a dark corner of the basement, it may be too far away from the area Loki perceives to be her territory. Even when you can't smell the litter box, Loki can and its smell reinforces her claim on her territory. If she thinks that the area is challenged at all, even by you or your dog, she may avoid it. Cats are extremely sensitive to territorial disputes and most will do anything to avoid conflict, including not using the litter box. If it's possible for you to place the box somewhere where your dog can't get to it, that will be best.
You may want to address the kind of litter you're using but if she's using it part of the time, she must be recognizing it. Most cats prefer sandy, soft grains to some of the newer pellets and chunkier textures.
The fact of going outside the litter box actually introduces additional stress to the situation because Loki would rather go in a place where she can cover her waste and feel good about it. One thing you can do is to treat her when you notice she HAS used the litter box. Don't hover - no cat likes to be watched - but if you hear her scratching around in her box, a treat is in order.
Overall, the trick is to make her feel happy and safe in her new home. Make sure she has plenty of high spaces to climb to - most cats feel safest up high - and make sure she has at least one spot where no one, not even you and especially not your dog, can reach her. Don't pressure her or punish her when she goes outside the box. Punishment doesn't work with cats, so no harsh voices. Only kindness and understanding will solve this issue, and you've already illustrated both by reaching out about this. Thank you!
We've had some similar questions in the past. I'll include links to those answers below in case something there is helpful as well.
It may take a little bit of trial and error to figure out just what's bothering Loki, but I feel sure you can sort it out as long as you address it kindly from Loki's perspective. We wish you both all the best!