I dont know what to suggest re your dog. Especially as I don't know no anything about to 'treat' or train stray dogs.
But wrt to her pushing in between you and ds2, I do think you need to how her that she is just a dog, and people come first. IMO that also means that she shouldn't be allowed on e.g. the sofa at all times. Being allowed to be on your lap on the sofa at some times, but that being pushed away at another time would make no sense to a dog. I always think dogs are very autistic in their thinking. All rules need be be incredibly clear and strict.
Our dog does get lots of cuddles. And he does sit or lie on my lap. But only when I first sit down in the floor and invite him on my lap. As we had him from tiny, it was of course easier to enforce it, as he had never been allowed on the sofa before. But there would be quite simple methods of teaching her. I've done it before with my previous dog, who was lovely but quite dominant.
Another 'tool' we have used to teach our dog where he is allowed or not allowed , are stair gates. He is not allowed upstairs at all. He only tried to go upstairs for the first weeks, then he accepted that his place is downstairs. The stair gate has been removed quite some time ago, as he wouldn't try to come upstairs anyway. We also have a stair gate in the doorway to the kitchen. He is allowed in the kitchen (but he never steals food), even at mealtimes. As long as he lies down quietly. If he begs for food he will go out and the stair gate is shut.
And he never ever gets human food. And with everything we do or give him, it is always clear that he is 'only' a dog, yet a very loved one.
Wrt the barking, the only thing I can think of is really rewarding her whenever she is quiet, especially when adults around. But as you said that she has such a sensitive stomach, I'm not really sure how you can reward her, apart from praise as well. You really need a professional for help. Shame on that charity leaving you along like that!!!
Anyway, kids running riot. I am I have not one across as patronizing as I really don't mean to be!