Well, I have some good news and some bad news. The good news is we've been there - and saw a HUGE difference. You can change the behaviours.
The bad news is you're likely causing them. From what you've written I can see a few things happening that would cause the behaviour.
I highly recommend looking up Dr. Ian Dunbar - his website has tons of info for how to train your puppy using dog behaviour - not alpha stuff - that alpha dog stuff tends to cause more problems than it fixes.
In Jan we rescued a large bree d puppy - at 4 mos he was 50lbs (at 11 mos he's 90lbs) he was a sweet puppy, but he jumped, he growled, he lunged, he nipped, and he even bit me a couple time (not enough to draw blood, but he left bruises).
First, you need to train your puppy - not the kids, once he's more predictable, then get the kids to help. Second - do NOT play tug-of-war. Tug-of-war is a game that establishes dominance. Until your puppy will drop whatever you have as soon as you tell him, do not play this game. it DOES create aggression - at the very least it lets your dog believe he's boss.
Also, if you're not already crate training - do so. It'll really make a huge difference. If he's not behaving crate him. If you can't watch 100% crate him. Get a few kong toys - he gets his kibble from toys or hands only. Not a dish. This rewards him for playing with his toys - not the kids toys.
When giving treats by hand - close your fist - he does not get the treat until he stops touching your fist. At first he will lick, and paw at your hand - as soon as he takes his nose away from your hand open your hand - after he starts to get the idea, then keep your hand closed until he doesn't touch you for 10 seconds, then 15 then only if he doesn't try touching you at all. If he jumps, turn your back and cross your arms. Don't say anything, just turn around give him a treat when he puts all 4 feet on the floor.
To inhibit bite - grab his mouth and force your hand in - so you're circling his lower jaw with your hand - he will push your hand out - when he does reward him with a treat.
Teach him to sit, lie down, roll over. If he's sitting, he's not jumping, if he's lying down he's not jumping. If you don't know anything else to do - make him sit/lie down and reward him for that vs accidentally reward him for a bad behaviour.
Attention is reward - so if you yell at him - it's a reward.
If he's gotten a toy, chewwed a shoe - it was your fault for leaving it out - do NOT pull it away from him. Everytime you pull it away from him gives him information. It lets him know he must defend everything from you b/c you steal things from him. Teach him drop-it/off/leave-it some command that lets him know not to touch. Everytime he drops what he has, reward him - it rewards the drop - he will eventually only get his toys, then you can move on to games like tug-of-war where the game is enough reward to drop-it and get-it.
anyhow - this is basic information - I am not a dog behaviourist - but this is what's worked for us. I think you should (at the very least) ditch the trainer you have - but better yet find an actual dog behaviourist to help you out. but look at the credentials - you want someone who actually has the education to back up their claim to the title.