I agree 100% with Heather. I think a big part of the problem is that your cat is stressed out about the box...probably because the other cat or the dogs are keeping her from feeling safe when she goes to pee.
The first thing I would do is to get her her very own litter box and put it in a place where she can use it without being surprised by the other cat or the dogs. Make sure there is a little privacy but that she can see whether another critter is sneaking up on her. A cat will not use the box if she is afraid she will be attacked while on the box!
Then...like Heather said...you may need to separate the cats for a week or two. Lock one in a room while the other is out. Don't let both out at the same time for at least a week. Then, when you do let them both out, give them a little tuna water or tuna so that they associate the other cat with a pleasurable event. Give lots of treats whenever the two are together in a peaceful way.
Also, like Heather said, make sure you thoroughly clean all old urine spots. To find them use a flourescent black light (a regular black light bulb won't work) which will make the old urine (and barf) glow. Saturate old spots with an enzymatic cleaner.
If she tends to pee in the same spots, you can lay strips of aluminum foil over the old places which will keep her from using them.
Also...explore using different types of litter. Many cats won't use scented litter and most prefer the fine grained clumping type.
If you use a covered litter box, remove the lid...many cats refuse to use covered boxes.
Please have her checked for a urinary tract problem. Even when sick they will not avoid the litter box every time. When my cat had the urinary tract problem she only peed outside the box once or twice a week! Once her physical problem cleared up she never peed on the rug again!