I imagine there will be some tussling so they can figure out who's going to be the boss between the two of them...I wish I could offer expert advice. I'm sure you'll bet some good advice from the folks on the site who are good at behavior stuff.
Hard to respond without knowing more. Usually opposite sex dogs are less inclined to fight. What was the fight about? How did it start? What did you do? Difference in size should not be an issue. BTW it's not HIS turf, it's YOUR turf and you get to set the rules. If you think of it as "his turf" you will be more inclined to excuse behavior which you are not happy about.
I wouldn't allow either of them on your lap for now. It sounds like they were resource guarding you or the bone, or a combination of the two.
Did the breeder specifically say she was very submissive before you got her?
Google Nothing In Life Is Free and follow it with both dogs. You are the leader and get to decide who lives in your house. They are trying to decide who is in charge but are forgetting that you are!
Debra, I would keep them on the floor, if this is an option. When you have dogs on beds and sofas with you, it aggravates any dominance issues. If you choose to allow it, at least be aware of this. As Jane said, I would also keep them off me. You need to be the clear pack leader, you can get back to being more of a buddy when the situation is well under control. As for toys, do away with any edible type toys ( rawhides, real bones, chew hoofs, bully sticks, antlers... the list has become so long.... ) stick to the toys that are more conventional and obviously not toys you put food in! If a toy seems to create more of a problem, do away with it for awhile.
If, in spite of this, you see a squabble starting ( or even just posturing ) a loud "Knock it off" should return the upper hand to you. The mssage needs to be clear that this is totally unacceptable to you and you are the one who can make it stick! Keep the dogs separated when you are not around, until the relayiionship is well established and old hat. It should not take too long, weeks not months. Long walks on leash together are the best way to socialize them to each other. Having a clear routine of what happens when, where and in which order, will also make for an easier adjustment, as the dogs learn what to expect in this new three way situation.
I agree completely. No high value toys/treats til the are used to each other. My rule in my house is only one dog is allowed on my lap at a time. I generally allow the one who had first dibs stay there, but if that one gets snarky (in this case, that would be Jack) he immediately gets put down. However, I minimize this by not allowing Madison to jump up if he's already there. The one thing he won't tolerate from another dog is crowding.
Once mine started to escalate into what sounded like genuine anger to me, and I said "Everybody QUIT IT!" and they both looked at me like "Huh? Oh." and that was that.
When I brought in my new one, I kept them separated for long stretches and brought them together for short ones, gradually adding length as the weeks went by. Newbie gets confined a lot. Them's the rules.