Absolutely not. We've had no tributary relationship with the British crown since the revolutionary war ended in 1782. American attorneys also don't wear wigs in court. Where did you get this idea?

That's a crazy conspiracy theory. I guess I could write a whole long post about why it's stupid and wrong, but I'm not sure it's worth the time. I'll just say a quick piece, and also mention that that which is asserted without evidence can be dismissed without evidence.

The word "bar" comes from an actual physical bar that was in medieval courtrooms and separated the people specifically dealing with the case legally, from the general people attending the trial. Medieval courtrooms were kind of like crazy free-for-alls, with attendees shouting out things, and angry people shouting, and people crying and trying to get to the accused, that kind of thing. So there was a bar audience members were not allowed to cross. On the other side were only lawyers, the accused, bailiffs, and the judge. When there began to be a test you had to take to be a lawyer, if you passed the test, you gained the authority to be on the other side of the bar. You were "admitted to the bar" and you were a barrister and a member of the bar association. This word has been around before the New World was even discovered. For reference, please look at the Oxford English Dictionary definition of "bar"

Crazy conspiracy theorists are constantly looking to find "the truth" through hidden meanings and symbols. So they came up with this thing that "bar" is really an acronym (or, more accurately, initialism) for British Accreditation Research. B.A.R., get it!? No, it's not. They made it up, but then they can tell people how clever they are for knowing this thing nobody else knows, that proves that the US is really a British colony and secretly controlled by the Queen and the illuminati, etc. No. That is all in their imagination.