What type of Boxer conforms to dog show standards?
It depends which country you are in. In America, the judges tend to base their opinions on not only the American breed standard, but how the dog compares to others most commonly seen at shows. The written breed standards for many clubs can be applied to all 3 styles of Boxers.
Some will say that U.S. judges tend to favor American Boxers and European judges favor German Boxers, however this is up for debate.
Do breeders ever mix bloodlines?
Yes. Some breeders who live by the rule that a Boxer is a Boxer, will mix bloodlines. Others hold a strict standard to never mix at all and work to preserve a strong line without any other types being mixed in. An owner may very well decide to have a German/American Boxer or other mixed type. Many dogs have both American and European bloodlines...
And with all of the breeding that has occurred over all of the past generations, one would be hard pressed to locate a dog that does not hold a trace of 2 or even 3 of the different bloodlines somewhere, even if it is very far back in the pedigree.
Which type is the best?
Generally, temperament is the same across the board. There are some that will say one type has more energy than the other, or one bloodline has more aggressive tendencies, however this has nothing to do with UK vs US or anything of the sort, it only has to do with the particular dog.
There is no one to judge which Boxer breed type is the best; it is all a matter of personal taste. Someone living in America may have a gorgeous German Boxer and visa-versa.