Big dog Little dog -- who's the dominant one?

I've been looking around the forum and trying to learn as much as I can,as quickly as possible. I foolishly brought home an Australian Cattle Dog/GS mix from the animal shelter a week ago today. She's beautiful and really smart, but she has a ton of behavioural issues, mostly of the separation anxiety type. I had a border collie/shepherd mix for ten years -- she passed away last fall from a brain tumor. So Janey's characteristics (the new dog) are somewhat familiar, though she is a) way more energetic than Emma was, and b) far friendlier with people and other dogs.

One immediate problem is with our other dog, Kara, a pekinese/chihuahua/spaniel mix (ugly, but sweet as can be). She's a lap dog, very attached to her people. She tends to be aggressive toward other dogs, especially bigger ones! If Janey gets too familiar with her, tries to sniff her, gets too close, Kara bares her teeth and growls pretty vigorously. Then Janey reacts by getting very excited, poking at her, dancing around her, yelping and whining (herding behavior?). Kara stands very still, keeps her head up and will eventually strike. Earlier this evening such an encounter happened and resulted in a dog fight, with Janey pinning Kara to the floor. No injuries, but Kara was pretty shaken up. I didn't see all of it, but it looked to me like Janey was not biting. We shoved Janey outside and made her stay out for a bit, then let her in on the leash. She's tied up right now and behaving herself. Janey also gets very agitated, yelping and whining, when we pay attention to Kara, or when she sees her up on our laps. Her previous owners apparently did let her up on the couch and on their laps -- she weighs about 40 pounds! She acts like a dog who's been in a one dog, doting permissive household with no discipline.

When I read the information about introducing a new dog to the house it was suggested that we need to figure out who is the dominant and who is the submissive, and then reinforce those roles. But I'm just not sure. Janey keeps doing stuff like trying to take over Kara's bed, but only when Kara's not around. If Kara is on the bed, Janey stays away. (it's a big bed meant for the old dog, who would have nothing to do with it, so Kara took it over). So my sense of things is that Kara's the dominant, Janey's submissive, but Janey is really trying to push Kara's buttons -- perhaps I'm seeing herding behavior, rather than aggression? I'm worried about Kara getting hurt. My sense is that we need to keep reinforcing us as the alphas and keep both dogs in line, reprimand Kara for growling and Janey for acting up. But that's different from the approach in the book.

By the way, we're walking both dogs twice a day for at least 30 minutes. They get along just fine when out on their walks Janey pulls but is learning. Kara gets an extendable leash and usually leads the way. They get along fine in the yard (my husband and I are the problem!) Working on sit, stay, down, and wait with Janey. She learns fast, but is stubborn, persistent, and sensitive. I think I'm doing all the alpha dog stuff you've listed food, doors, etc. She also chases cats, jumps brick walls She'll be a great dog someday!

Sorry for the long post! Thanks for any help you all can provide!