I don't think that breed affect how a dog should be trained nearly as much as the attitude of the owner. The more the owner hides behind a breed label the less a dog will learn.
I don't have what is thought of as an "easy" breed. My dog is bred to hunt independently, he is not bred for focus and close control but because I keep an open mind and try out suggestions I have learned to train him better and better and he is now more focussed on me than a lot of border collies.
American bulldogs and pitbull terriers (similar type to Piper) are not bred for focus and close control either. If Piper was a border collie you *might* be able to hold that point of view. As it is
Lotta's breed has a reputation for being easy to train and for being very focussed on their owners because they like food, and
in this link the Cavaliers are listed as easy to train and Pit bulls are listed as hard to train.
What we wrote about breeds is based on what my friend knows and what she has read/heard. She has read that some breeds are easier to train than others. When my friend has seen dogs on TV and in videos that are easy to train, they have often been border collies, german shepherds etc. It seems to my friend that those breeds just naturally know how to work with their owners. She hasn't seen Cavalier's for example in tutorials. Those Cavalier's she has seen in videos, do silly things like Lotta does. Although you are saying that they are easy to train, Lotta is more similar to those breeds that are more independent. Things should be done differently with them.
So, we have told that Lotta gets excited when she notices that my friend is going to start training her. Lotta behaves the same way also after my friend has taken her out or she has been playing with her toys alone or with my friend. So, doing something else first doesn't get Lotta to calm down for training.
I feel like you haven't tried any of the suggestions given to you. Even if you feel like they may not work you should at least try them.
Such as I suggested you tire Lotta out before training and it may help her focus on you.
Of course my friend could try what you have suggested. The question is how could she apply them since we have told what my friend can/could do and what she can't/couldn't do and what kind of limitations she has:
"We have told that my friend lives in a small apartment where there isn't an extra room and that she doesn't have own yard."
"If Lotta noticed that my friend would start gathering the toys
she would get even more excited about them and would try to take them from my friend's hands. Although my friend would manage to pick the toys up
Lotta would get frustrated about it. Then she wouldn't be in suitable mood for training."
I really think you need to accept that Southerngirl is able to train Piper because Southerngirl is a very good trainer. It would be a good idea for you to take her advice.
Dogs wander off in training because you are not keeping their attention. You need to train for less time and/or increase the rate/value of rewards. It is as simple as that. If this works for stopping my dog from chasing squirrels during training sessions then it will work to stop your dog running after toys during training sessions.
Lotta doesn't always run away in training but when she does she does it even when my friend has gotten her to do something several times and has rewarded her many times like in our video we had linked. The lenght of the training session doesn't affect in it.
So how could my friend teach Lotta to do same things as those dogs do in that video?
For example in the following video those dogs concentrate on doing what that trainer asks them to do no matter what ever is happening around them because they have been teached to do so:
They don't break the eye contact or wander of even when that trainer throws toys on the floor.
My friend knows that one of the reasons those dogs can do those things is that they have good impulse control. Does someone know what else that trainer might have done so that he had gotten them to work with him?
So, you had suggested It's yer choice. How could my friend teach it and those things you had told here since Lotta doesn't know how to take and drop on cue yet? How could my friend prevent Lotta from stealing the toy then?:
Using the distraction as a the reward is not the same as using play as the reward. Being the gateway to the distraction works in 4 ways and the reward aspect is only one of them.
1) you remove conflict by permitting the dog to do what it wants to do
2) you make the dog think you have control over what it wants to do (very sneaky ) .
3) you make the distraction less appealing by encouraging the dog to go and enjoy it.
4) you reward the dog with what it wants.
By the way, why do you think you can only use play as a reward after teaching take it and drop?
You had also suggested to use LAT. How could my friend do it with toys? It's also so that all of those things would be easier to do with animals because they are fast enough to run away. So, it's not so clear to my friend how to do them with toys.