1. Crate training: I do have a fairly big garden and not the kind that is reachable from the road so stealing the puppy won't be an issue but I still would feel safer if the dog is properly crate trained as it will help me for transport and I somehow prefer to have him inside at nights
Absolutely crate train your pup! That is the way to go. A dog kept outside often turns into a nuisance barker and becomes destructive. Much safer inside. Being free in the back-yard allows the pup to develop bad habits like digging, and the time does not count as exercise. Pup will still need to be exercised and played with when people are home, so it is not a 'short-cut' for the time a puppy or older dog requires.
2. Biting: I know a bit about soft mouth but I am concerned about the kids would a puppy's bit be serious enough to hurt them before he is used to "soft mouth" ?
The concept of 'soft mouth' if pretty much lost on a GSD. They aren't called landsharks for nothing. Yes, they can inflict pain and damage in play-biting, but it is a phase. The whole family, including the children will need some training on how to properly play with the pup, chief thing being redirecting those puppy bites onto toys.
3. Training: I am planning on train the dog myself, nothing world class or competition, just obedience training and, above all, soft temperament, I don't want to end with a dog as a liability. I am not convinced in any training center we have here
Nothing wrong with that if you feel confident that you have the skills and experience to do the training yourself. Just don't neglect socializing your pup to the big huge world and proofing his obedience in new places with new distractions.
4. Leash walking: loose leash walking is very important for me. Since we are a family and anyone could take the dog for a walk and an adult GSD can rapidly out-strong us all
My only concern is that I wouldn't expect a 10 year old child to be responsible for a GSD on a walk, no matter how well-trained.
5. Separation anxiety
Best bet is to fine a breeder that has a track record for breeding solid, confident dogs. Get references, talk to the owners of dogs from the breeder, see if there is a pattern of separation anxiety, or general shyness and fearfulness.
6. Socialization: how to do it correctly and what does it include? only people or also dogs/cats/cars/noise etc...
To me, socialization is EVERYTHING. New people, animals, noises, places, outings, stuff. I live in the boonies, and prefer a quiet, reclusive life, but in the first year of having Gryff, I made regular efforts to drive into town for socialization walks. We'd walk around mall parking lots, go to a ski hill, check out horse arena's, watch a hockey game, hang out with skate-boarders, etc, etc, etc.
Socialization does not mean that they have to interact with every single person that they see - just that they do see lots of people, and know that all those different people are 'normal'.