I heard some theories that if a cat or dog is abandoned, when it is adopted again, it is harder to train them because they may be unable to trust human anymore, is it real?
I caught a wild cat of obviously unknown history. She had to spend 10 days at the animal shelter before I was allowed to adopt her. She was temporarily placed in a foster home to be socialised and was evaluted as being unfriendly and anti-social at the end of her stay.
When I got her after the 10 days I spent one and a half hours with her in a single room with the door closed every day so she would get used to me. Most of this time was spent with me sitting on the floor and her ignoring me.
Eventually she began to trust that I wouldn't harm her and she started to interact with me. Two years later, she now comes and sits on my lap, she loves attention, she asks to be picked up and cuddled and she has integrated with my other two cats.
My other two cats were originally feral kittens. All three were originally little spitballs of pure hate and fear but now all three of them are totally beautiful cats! I did exactly the same process with the kittens as I did for the adult cat and, with patience, I got the same result: a loving cat in each case.
So it might be hard, but it certainly isn't impossible. The important thing is to act in a way that is consistant and that they can trust. If you break their trust you could permanently damage your relationship with them.
I only know dogs.
A lot depends on what they went though. Most dogs can be rehabilitated (learn to trust). You can get an abused dog that may never come around.
I do Monday - Friday care with a foster agency and get the challenging dogs. By Thursday they typically bond. But I had one girl that must have been abused by a man because, even after 4 weeks, I got no where with her. However, she was fine with all the women at the agency.
Establish yourself as the alpha can be hard. If you get a defiant dog with trust issues, then you cannot even begin training. If they are destructive, then it can be a real problem.
Once you bond, it can be even deeper than if you had them since a puppy. I have a girl I adopted that will not leave the yard, even if the gate is open.
Get a dog from a responsible shelter that will help match you with a dog.
Once you have built trust, then training is just basic dog training. It will depend on the intelligence of the dog and your ability to train.