There's a lot in here. Let's break it down.
This is a common mistake that is made with socialisation. It shouldn't
be about meeting a ton of dogs, it should be about observing (many things, not just dogs) without interacting. What may have happened is that you have flooded her and now she is no longer a pup, she is telling other dogs to get out of her space.
This is actually not uncommon but very few dogs really want to get into a fight. All of their instincts tell them not to - in the wild, the risk of injury is simply too great. In fact, aggressive behaviour is almost always rooted in fear.
By putting on a big display, your dog is trying to frighten off the other dog, her body language is saying 'I'm loud and big and scary, don't come close to me if you know what's good for you'. And almost always the other dog will retreat, or be taken away by his owner, so your dog's behaviour becomes reinforced. It worked, so she knows she can do it again.
This sort of behaviour often happens when your dog is on lead, which means that she has found herself closer to the other dog than she would have chosen if she had been able to.
She will have an invisible radius of space around her where she feels secure. It's called flight distance, anything within that space triggers her fight or flight stress response, which you may have heard of. Find out what that is and keep her far enough away from other dogs that she is aware of them, but relaxed. Your goal is to train that she doesn't need to react; not to stop a reaction in progress.
Reward her for being calm with something fabulous, like frankfurter sausage or a very special toy. The aim of this is to change your dog’s emotional response to the stressful thing (the other dog) by repeatedly pairing it with something good. In time, your dog will learn that scary dogs mean sausages appear and this creates something called a positive conditioned emotional response (+CER).
This website explains it in more detail - Care for Reactive Dogs
Gradually, over weeks and months rather than days, you can work on reducing the distance. This may mean you have to be selective where you walk - choose places with good visibility so you can give other dogs a wide berth, or where you can turn and walk away easily. But - be aware that if your dog has had a stressful episode the stress hormone cortisol can stay in the body for some time. Studies in dogs are inconclusive but it may be several days. That means that if her cortisol level is already high, the distance she was comfortable with on one day might be too close on another day. So the safe distance can change, watch her body language.
Alongside that you could train a 'watch me'. As your dog looks at you, mark and reward the behaviour. Ask for longer periods of watching. Then if a dog approaches, after you have worked on the distance issue, you can get your dog to focus on you and not the other dog. BUT - some dogs find this scary as they cannot see the thing they are anxious about so you need to judge your dog. And importantly, don't ask your dog to watch you if it is the other dog that is reactive. Your dog should never be in a situation where she could be at risk while she is complying with something you have asked her to do.
Trainers describe behaviour like this with reference to the three Ds. Distance, as above but also be aware of Duration - your dog might be tolerant for 10 seconds, but not 15; and Distraction - how distracting the stimulus is, a calm dog might not trigger any reaction at a given distance but a bouncy one might.
In addition, the conformation (shape) or even colour of some dogs can trigger a reaction. Very broad fronted dogs (such as mastiffs or bulldogs) create the impression of 'facing up' just because of their shape, which can be intimidating even if their temperament is perfect. And black dogs are thought to have facial body language that is harder to read. Some dogs will be more reactive to un-neutered males, or particular breeds for no apparent reason. Learn what triggers reactions in your dog so that you can give her the extra support she needs.
Harness and leash
First make sure nothing is uncomfortable. Nothing rubbing, no buckles or fasteners near her elbows etc. Is it a Y or H shaped harness at the front? H shaped harnesses, where a strap goes horizontally across the chest, restrict free shoulder movement and cause orthopaedic problems. Some dogs don't like harnesses that have to go over their head, or that have fasteners that make a loud click. If any of that applies, change your kit.
Then, slowly build up her confidence with it. Put it on the floor. If she looks at it, reward (immediately so it's clear what the reward is for). A clicker or other sound marker would be useful for this, so as not to derail this thread there's one here on how to use a clicker Behavior Diagnosis
Build up slowly, to her coming to you to see the harness, touching the harness, holding it against her side etc. until you can put it on, using lots and lots of reward
Pulling and Recall
We have a whole area of the forum dedicated to recall and walking nicely on leash. I'd just add, it helps you if you look at the work you put into this as a training exercise as opposed to trying to go for a walk. Not having the idea of actually going somewhere takes the pressure off you.
Treats and the other dog
Resources and guides to train loose leash walking and recall
Animals are hard wired not to share. In the wild, it could be the difference between survival and death. So never give the two treats together.
Do, however, by hyper-aware that this could transfer to your child so be very vigilant. If your child is eating, I'd suggest simultaneously feeding your dog but separated by a baby gate.
This is NOT a step to ensuring a safer and happier relationship.
If you were in a restaurant enjoying a favourite meal, and someone came and snatched your fork out of your mouth and took away your food, how do you think you would feel? If they did it repeatedly, you would (a) start to get more nervous that your food was at risk; (b) start to get more possessive and guard it harder; and (c) get creative with your cutlery - your dog doesn't use cutlery so she might use her teeth.
It's not a bad idea to teach a 'drop it' in case she gets something she shouldn't but if you have to ask her to give something, always
have something better as a swap.
Picking her up
Pretend she is a Great Dane. Honestly, bear with me.
We pick up small dogs because it's convenient for us but like in the analogy I gave about food, how do you think you would feel if you were dozing in a comfortable chair and someone suddenly and unexpectedly threw you off it? You wouldn't be able to just lift up a Great Dane, so treat your dog in the same way. Instead of lifting her, lure her off using a treat and at the same time giving her a verbal cue like ”move off”. Once she understands, you can use the cue and treat her as a reward for complying.
That said, there may be times you need to lift her, for example on to the vet's table - so train her to accept being lifted.
First, do it when she is awake, not dozing in the sun. Then, tell her so it isn't a surprise. Say something like ”lift up” and use one hand under her chest and the other supporting her hind feet. Immediately reward, using your third hand (sorry
) and put her down. Repeat regularly.