First of all, you’re pup is ADORABLE and I really commend you for going to such lengths to help him overcome his crappy start in life.
Second of all — please please please do NOT NOT NOT use dominance methods with this dog. I am not opposed entirely to all dominance methods but this is definitely 100% not a situation where they are suitable.
Third of all, your dog came from a puppy mill which means that he wasn’t only not socialized to human life, but he missed out on crucial socialization with doggy-language too. This is very important to keep in mind especially with the resource guarding (and conflicts with your female).
I’ll go through each “issue” in turn: 1) Barking at kids; 2) running away down the road; 3) resource guarding.
1) Barking at Kids
I reposted this particular picture so that I can outline a couple important things. One, notice how tucked your dog’s tail is at the top — highlighting that he is tense and uncertain. Second, notice that even in this crouched position (I assume she was askig him to lie down) he’s trying hard to keep his weight back away from the child while still getting the treat. Three, notice how the little girl has her body directly facing your dog head on and is crouched over your dog’s head/front. The first two are signs of how nervous your dog is about this situation, the third one is a very good reason WHY he’s nervous. Dogs, especially nervous dogs, perceive head-on approaches as quite aggressive and being crouched over is very threatening for them. So it’s very understandable why your dog is showing real nervousness in his body language, however he’s working hard to perform for the treat – what a good boy!
I would put a pause on direct interaction with children right now and work your way up to that. Instead, have your dog on a long leash (10 feet minimum) and slowly approach kids in the distance — what that distance is is up to your dog. You want to get him to the point where he’s aware and focused on the kids but only showing minor signs of uncertainty (lip licking, yawning, ears perked/focused eye contact on the kids). At this point just hang out, don’t move forward and just let your dog mosy as he wants on the 10 feet lead (don’t let it drag so that he gets tangled…just let it hang loosely without touching the ground). Have some treats, but don’t feed them to him directly, toss them around you so that he has to sniff them out. Sniffing self-soothes dogs so it helps him calm himself in the presence of “scary kids”. This works best on grass (so that he HAS to use his nose rather than just being able to see the treat on pavement). If your dog ignores the children and sniffs around (not treats), just let him. Don’t follow him backwards/sideways, but if he follows his nose forward toward the kids then follow slowly. The key is letting him set the pace, using treats to engage his nose, etc.
During this exercise you can also do some intermittent obedience practice (sit, paw, lie down) but try to use treats he needs to lick (e.g. cream cheese) rather then hard treats as licking is another self-soothing action for dogs.
As you build up and he gets more confident and relaxed with kids being close to him (but make sure they ignore him) you can begin working on interactions with kids again. When you do this, don’t have the kid ask him to perform tricks. Just have the kid standing sideways to Tulo, have a small tupperware or similar with cream cheese (or similar) spread thinly around the edges, and just let Tulo walk up on his own volition to lick clean the tupperware. The kid should just stand staring forward (not looking at or trying to pet Tulo) and let Tulo approach, eat, and decide when to leave the kid. You can build up slowly from this over days/weeks to the kid facing Tulo but not engaging, the kid glancing at Tulo as he approaches, the kid making eye contact occasionally, the kid petting Tulo, etc.
2) Running away – long line (50 ft). Nough said. He’s a pup, he doesn’t know to stay near you. This doesn’t mean he’ll ALWAYS need to be on lead (so don’t despair) but I would recommend he stay on lead in unfenced areas until he doesn’t test the lead’s boundaries.
3) Resource guarding. PP had a good recommendation (tied up, waiting to relax, etc). When you can’t do those exercises, you just have to manage the situation by keepign him separated (crating works well here). This doesn’t mean you separate him once he acts badly, it means he’s separated FROM THE ONSET until he understands food = relaxed.
Your female is absolutely acting appropriately by noticing his poor manners and saying “hey, cut it out” — Tulo is acting badly by responding with aggression/assertion. This is just a matter of Tulo not learning proper doggy manners/language as a pup, so it’s not his fault and he ‘s not a bad dog. However he is what he is so now you just need to manage the situation. Don’t put him in situations where you know he’s going to become possessive and not be able to communicate properly with other dogs, since each time he has this type of interaction it entrenches the habit. So prevent that habit from further solidifying by just keeping him removed whenever food comes into the equation.
This sort of issue is obviously best done with atrainer, but keep heart that he is definitely not a lost cause! However he will require really diligent and consistent handling to help him overcome his puppy mill past and become more confident and relaxed in his “new” environment!
ETA: When working with kids keep in mind that working on it in multiple short bursts is better then spending LONG periods working on it. It may not seem it, but he’s likely to be in aheightened state during these sessions which can be incredibly draining and stressful. Imagine walking home in a strange neighborhood at night and there’s a random guy walking behind you — that’s the same level of fear that he’s feeling so keep the sessions short and positive so that he has time to “come down” from that level of alertness.
Also — when you get to the point of working on direct interactiosn wtih kids, I wasn’t clear that he child is holding the tupperware. You can also start out with the child sitting cross legged holding the tupperware out to their side and have the dog approach the same direction (i.e. chld is sitting with their side facing the dog).