1. she can sometimes bark at a larger, strange dog -- I guess this is fear
2. If there is a stranger dog to the pack then she plays well, but if that stranger, on the first day, tries to take the stick she is gnawing, she fights. She doesn't fight with her friends if they take her stick and she doesn't fight with stranger dogs if they don't
I'm guessing this is normal doggie activity -- is that correct?
Also, what should I do if my dog is fighting with a dog that is more or less the same size? If it is a larger dog, I try to get the owner to do something before mine is hurt and if it is a smaller dog (which hasn't happened, but she can sometimes play too rough with say a yorkie) then I put her on the leash and sometimes put her on her back for the littler dog to smell her and she's ok with that.
But if they're the same size, one "group of thought" says "let them fight until one gives up, then if the other doesn't back off, then intervene" -- and I'm inclined to agree with that -- is that correct?
Do you watch Cesar Milan? You'll notice that Cesar is always the pack leader. He doesn't let the dogs decide these things. Without seeing how you react, I'm not encouraging or discouraging what you currently do. But I'll tell you what I do and what I recommend to people.
Our dogs take their cues from us. So if you allow your dog to fight, they understand that means fighting is ok, or even encouraged. If you are ok with your dog fighting, knowing that (a) your dog might get mortally wounded in the process, or (b) your dog might mortally wound another dog... But the person who owns the other dog might not appreciate it.
I don't let dogs fight. Ever. I stop it before it starts if possible. If a fight has started, I break it up immediately, and both dogs get very sternly scolded (unless one obviously was trying NOT to fight).
In the example you gave, where you say you let the other dog's owner do something to stop the fight: I never do that. I don't care who's dog it is, if either dog starts posturing to dominate, or growling, or eyeing the other dog, or curling a lip, etc., I tell them in a very stern and DEEP voice, “No! Knock it off!” I make it very clear that that kind of behavior will NOT be tolerated.
Dogs who growl and posture and fight (at dog parks, for example), are dogs who's owners have allowed them to do that behavior, and it will escalate. Watch some afternoon Judge shows on TV, and every other story is about dogs, and the root of every problem is an owner who allowed their dog to do these bad behaviors until it finally resulted in a tragedy ($5000 vet bills for the neighbor's dog, or the death of the Yorkie, etc.). Those dogs will always be bullies. That behavior is no more appropriate in our dogs than it is in us. And of course, there is often a correlation between the owner's tendency to be a bully and the dog behaving like that. The dog has been rewarded for that behavior.
It's one thing if two dogs are friends, like you said, and they are PLAYING, and in the course of playing, they are FRIENDLY growling and PLAY fighting — i.e., rough-housing and wrestling. It's another thing entirely if one dog is truly trying to fight with another dog. You have to be able to distinguish between the two. You have to learn dog body language, dog vocalizations, etc.
My two dogs wrestle with each other, they stand on hind legs, with front legs wrapped around each other’s shoulders, and get mouthy with each other, being very vocal the whole time. But they are playing, and I know they are playing, and neither one ever gets hurt. One of my neighbor's who isn't familiar with the different vocalizations of dogs thought they were fighting, and wondered why I wasn't breaking up the fight. She had no understanding that not every growl or vocalization means aggression.
Your puppy's behavior sounds perfectly normal in a dog pack. But she is still LEARNING what is acceptable behavior and what is unacceptable behavior. She is learning this from you. You are her pack leader. If you allow her to fight and “guard resources” (her stick) as a puppy, that behavior will only get worse as she gets older. You are reinforcing that behavior by allowing it. Puppies are like children (do you have kids?). They will always test the boundaries and see what they can get away with. If they are not corrected (firmly but fairly), they will push that boundary again and again. If I were you, I would nip that behavior in the bud immediately.
Resource guarding is guarding anything that the dog decides he/she “owns”. This can be a stick, toy, food, bone, bed, house, or person. If she is “resource guarding” with a stick, she may begin to resource guard with you — preventing anyone from coming near you, even trying to bite if someone comes near you. Incidentally, dogs who “resource guard” do not pass temperament tests when they are dumped at a shelter, and they can be euthanized because of it - they are deemed to be unadoptable because they may bite someone in the process of guarding their resource (food, toy, etc.).
You can also set your dog up for success, by never allowing her to have a stick when a stranger dog approaches. If she is happily playing with a stick among her friends, and you see a new dog approaching, go to your puppy and take the stick from her. She can't guard it if she doesn't have it. Tell her “give” or “mine” and then take the stick from her, and say in a very happy voice, “Good girl!!” and pet her. Put the stick away (inside your jacket) and begin doing obedience exercises with her to keep her mind on something else. Practice sit, down, come, etc., and reward her for focusing on you (by petting her and telling her good girl).
If I know my dog has an issue with something, then I intervene before the problem arises. For example, my dog also doesn't like new strange dogs. So if we are on a walk in a park and my dog is off leash, I am always looking around to see if any other dogs are approaching. If I see another dog, I call my dog and put her on leash until the other dog has passed, so that there can be no confrontation, no trouble. My dog is large, and some people are afraid of her simply because of her look (she is harmless).
So in your case, one thing you can do it anticipate these behaviors, and direct your dog to something else so she doesn't need to guard her stick when the new dog shows up. As you said, after a couple days of getting to know that new dog, it's no longer a problem, so she can have her stick back.
The best way to introduce dogs to each other is by both owners walking with the dogs on leash, not standing still. Standing in one place allows the dogs to focus on each other, and if either one is jealous or wanting to guard their owner, you have potential for problems. But if both owners walk, with dogs on leash, the dogs are busy sniffing new territory and doing dog-things, rather than standing around trying to guard their owners.
In dog body language, it is an attempt to dominate when one dog puts their head over the neck of the other dog, or when they raise their head up high and stiffen their whole body to look bigger. So when you have your dog on leash and it is meeting/greeting a new dog, if you see this body language in either dog, both dogs should be backed away from each other and the dominating dog should be told “Knock it off.” Be careful not to pull UP on the leash when you do this, or that will raise the head of your dog, and the other dog will take this as a sign of dominance (which your dog may not have meant), and the other dog may decide to take that challenge. So when you pull a dog away, pull it straight back, not up. Again, this is another reason it's best to be walking when two dogs are introduced. The owners can keep them at a safe distance until they have walked long enough to be comfortable with each other.
I'm not a big fan of putting a dog on its back for another dog (I may rarely use this between me and another dog, but only in certain cases). It's not your dog's fault the little dog is little. You can put your dog in a “down” (i.e., laying down) so that she isn't so big, but rolling a dog onto their back is putting them in an extremely submissive position. She doesn't need to be submissive to a little dog, she only needs to be neutral. And keep in mind that many owners of little dogs have allowed their little dogs to become bullies because they think it's “cute” when their little dog growls and bares its teeth and tries to be tough. It is NOT cute, it is just as inappropriate as when a big dog does it, and little dogs who do that are brats. So if you roll your dog onto her back, and a little dog who is a bully comes up, that little dog may see your dog as weak and always try to bite/fight with your dog. And it's very hard to stand up in court and say “the little dog started it” when your dog kills the little dog.
So I would simply let your dog be herself, without forcing her to roll on her back. Just be sure she is neutral and accepting of other dogs, not aggressive.
Lastly, to your first point, about her barking at larger, strange dogs. Yes, that is normal, and probably fear-based. You don't want to reinforce that fear. You also don't want to scold her, because she is in the process of developing her instincts for what needs to be barked at, and what doesn't. She is learning her role as part of the pack, and when she should guard/alert, and when she doesn't need to. So when she barks at a strange dog, in a calm voice, say “I got it, thank you.” Don't pet her, don't coddle her, don't say (in a happy/high-pitched tone) “it's ok baby”, etc. All of those things say GOOD GIRL and she takes it to be reinforcing that behavior. You only want to reinforce that behavior if she is RIGHT in her judgment about the thing she is barking at. Every unfamiliar dog doesn't need to be barked at. A creepy guy lurking around your house does need to be barked at.
This may qualify as the world's longest post. ;-) I hope it wasn't too confusing - I didn't have time to re-read it to make sure it all made sense.