I've raised 2 labs and 2 golden retrievers for Guide Dogs for the Blind and owned various small mixed-breed dogs The labs and the goldens were all bred at GDB in San Rafael, CA. (Incidentally, GDB is a wonderful organization that provides trained guide dogs free to qualified blind individuals in the US and Canada.) These puppies were weaned at 6 weeks, spent 2 weeks with their siblings in a weaning kennel during which time they were handled and socialized by volunteers. During some of this socialization, an adult female dog not their mother was present. They came to us at 8 weeks (2) 10 weeks (1) and 12 weeks (1). The variation in the age at which the puppies were sent to us as volunteer puppy raisers had to do with the dog's health, the number of pups needing placement with raisers, and our availability. So far I can tell, the time they left the kennel had nothing to do with how much they mouthed. All of them were mouthy until they got their permanent teeth at about 4 months. After this mouth behaviors gradually decreased, but the labs continued to be more mouthy into adulthood than the goldens, and they were also more driven to carry around toys in their mouths.
There are a couple of strategies to stop chewing and mouthing, and we have used them all either alone or in combination. The first is the application of a substance called bitter apple on objects that the dog likes to chew (the labs tended to want to take the leash in their mouth when they walked, which was not ok). I didn't find the taste-repellent very helpful with the labs. Their drive to mouth was very high and they were able to ignore the bad taste. It worked better with the goldens.
The second strategy is a short, sharp jerk (correction) on the collar whenever the dog puts its mouth on something that it should not. This starts as soon as the puppy begins being handled by people and it is most effective if you spend a lot of time handling the dog and if the dog is given a special bone that he is allowed to chew, because puppies have a strong drive to chew when they are teething. It takes the dog a while to understand this correction, and get the difference between their allowed bone and a human's fingers and arms, but eventually they learn to keep their mouths to themselves and use their approved chew toys.
The third strategy is to put your hand over firmly over the dog's nose, so that he can't breathe. This guarantees he will drop whatever is in his mouth, so that he can mouth breathe. Again, consistency and repetition are the keys. You can teach the "drop it" command this way too.
When dogs are a little older (6-8 months for the breeds I have raised), you can teach them a "that's enough" command and use it if they get mouthy when playing. "That's enough" is used for activities they are allowed to do (like play) but that you want them to stop.
In my dog experience, mouthiness in puppies is almost universal because of teething, and the degree of mouthiness and chewing depends largely on the breed and whether they are are given something they are allowed to chew, and are given enough exercise and mental stimulation so that they do not become bored. Terrier breeds, for example, have been bred to grab onto prey and hold on to it, so once they get something in their grip, they don't like to let go. To help downplay this tendency, you can avoid playing rope-tug games with them. Retriever breeds have been bred to bring birds and small game back intact and give it to their master, so they don't have as much tendency to lock on the way terrier breeds do.
And yes, you can tell mouth aggression from overenthusiastic play or teething. Mouth aggression is usually accompanied by other signs - hackles raised, lips back, stiff body, intense look, and growling (although some dogs also growl when they get carried away in play.) It's hard to explain with words, but dogs look different when they are being aggressive and if you hang out with dogs a lot, you can almost always tell when trouble is coming.
