Welcome to YT! I am so happy to hear that you decided to open your heart and home to a rescue pup. It is a wonderful thing to be able to do and sometimes as you can tell there can be issues.
You said that he has been in your home for 3 weeks now. They get comfortable in their new home and their issues start to come out. It is great that you have a trainer to help overcome any obstacles along the way.
It sound like your little guy is trying to show you and your family who is the boss and you need to let him know that it is you are and
not him
I would suggest that when you walk through the door and he begins to bark and act excited that you ignore him. If he is barking at you and your son I would continue into the house not looking down at him at all. If he brings you a toy ignore this as well. He is trying to control the situation and your family. You can even turn your back to him when he is behaving this way to further ignore him. When
you want to play with a toy then
you should initiated play. Do not allow him to initiate play because he is then controlling the situation and you and your family. When it is time to play you can also use the reward and treat. Good behaviors get a treat bad behaviors are ignored and then play is stopped.
You also said that he sleeps with the kids in their bed which could easily turn into a possessive behavior. As cute as he is I would have to say not to allow this either. He will begin to guard the kids in their beds. IMO, I would use the crate instead for at night. Only allow him on the furniture when you say it is alright. If he comes up when he is not wanted put him right back down. when he stays down treat and praise.
If you plan on taking him in the car I would not allow him out of either his carrier or keep him attached to a doggie car seat. If you can correct what is going on inside the house you will be able to also correct the behavior inside the car as well.
I would also keep a harness and leash on him inside the house if he continues to act aggressive. When he acts out, you need to be able to control the situation but picking up the leash and calling him away. When he follows your lead you need to then treat and praise.
I hope this is helpful and with a little bit of work on your entire families part he will learn your house rules. Sometimes when we have a little one that we know has had a rough start it is very easy to pamper too much allowing them to take total control.