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I feel it is a dominance issue at least in part, though most dogs don't eliminate where they sleep. However, she might eliminate where YOU sleep. The behavior you are seeing is partly why I advise clients to never let a dog sit on the couch or chairs and not sleep in bed with them. Humans have to be higher in the pack order than dogs for you to retain control over them and the alpha position. A dog thinks differently than a person. They feel if they are physically at the same height then they are equal or maybe even higher since they have teeth and can bite good.
As alpha, they will show their displeasure in many different ways. They may eliminate in what you consider "your space". They might also eliminate in areas where they do not want another animal to go. So you may be right in that the cats might be partially the cause.
Your first step is to get her off the furniture and onto the floor. Get her a bed just for her. I'd also suggest a crate for when you are not home. Attach a leash to her when she is around the house so you can easily remove her from the bed and furniture. She is likely not to take kindly on the new rules so a leash can avoid aggression.
I'd go ahead and start serious obedience training as well. Before you can get into classes, I am including links to a couple of other sites that teach some good methods of training. Be sure and read both. http://www.sibes.org/sdo/index.asp~m=article&name=sdo.id.A000232 http://www.luckydogs.info/pdf/Teaching-the-basic-commands.pdf
Training works best if you train at least 30 minutes a day (two 15 minute sessions). I would start making your dog work via the Nothing in life is free program (NILF). It is outlined below. http://www.pets.ca/articles/article-dog_nilf.htm
You should also have a fecal exam done and a urinalyses to definitely rule out a medical issue such as worms which with new kittens might be contributing to the situation. I hope this information is helpful to you.
The following site is helpful. Be sure and click on the link to the left on obedience. and links on subsequent pages leading to detailed instructions.