OP, you keep saying that people say they can’t hear your child, that’s what people say to be polite, I get the feeling that maybe English isn’t your first language and that the ‘typical British-ism’ that will be coming into play here will be going over your head.
People will be able to hear your child, no matter how quiet she is, she’s a 4 year old, they’re heavy footed, have no concept of an inside voice and like to play/dance/sing whatever, but as that is a normal part of everyday life, most level headed people will say that they never hear a peep.
Case in point was my neighbours (semi detached house) saying they never heard the baby crying, when they would definitely have been able to as we could hear them snore (both beds against the shared wall).
People will be able to hear your family and are probably irritated by the noise, but accept that it’s part and parcel of living in poorly designed shared housing. By getting yourself all in a tizzy over this you’re just making yourself look unhinged.
If they’re taking the dog out for the day, as you’ve stated several times, they’re being responsible owners, the dog will be appropriately exercised and entertained while out of the flat, be grateful that you’re not having to listen to a bored, destructive dog whining because they’re left home alone.
You’ve decided to move out, which is fine, that’s your prerogative, however it’s worth keeping in mind that you find animal noises offensive, so it’s worth looking for housing where you’re never going to come into contact with pets, which would mean a detached property.
We all have our limits and have to be aware of them, however you can only control yourself in this situation, you don’t know what has caused the people above to take on their pets, asking them to get rid of them, or force them to stay in one room is just as ridiculous as if someone was to ask you to do the same to your child.