I am sorry that you are facing this problem. I know all about the frustrations connected with it. And the dilemmas.

Lets start by asking how far did the medical treatment go?
-Was there a COMPLETE urinalysis? testing not only for blood, but crystals etc.
-Were there any blood tests? diabetes, kidney, even thyroid problems can lead to the need to urinate excessively. Sometimes the cat simply cannot reach the litter box, or associates the box with the pain from the UTI or crystals.
-x-ray, to see if bladder stones are present. My Rutherford had stones but NO crystals, and the irritaion they caused the bladder must have been painful.

Besides antibiotics, have you tried any medication? If he eliminates inappropriately due to stress, a medication that relieves stress might help.

Now, I can't help you keep the dog away from the cat poop, but I can say that 1 litter box for 2 cats is asking for trouble. Cat's don't know how to "wait in line", so if one is occupied, well, the next best spot might just be the floor.

Also, some cats simply will not pee and poop in the same box, so if there is already a poop in the box, then the floor will simply have to do.

I once considered having my PRECIOUS Y.Lee cat euthanized due to innapropriate peeing. I was 100% certain that it was a behavioral problem. The onset occured simultaneoously with the addition of a new cat to the home. Turns out he had diabetes, and could not hold his urine. Moral is = don't assume it can't be stopped. Both my Sterling and Jim are on amitriptylin to stop territorial spraying. It works about 95%, I'd say.

You may not be able to stop it 100%, but I hope you can control it to the extent that you can live with it. After all, we all have accidents, and we all make mistakes.

BEST! of luck.