Yes. It is very old-fashioned, but not incorrect. The subject of a second person imperative is normally omitted nowadays, but was commonly included in the past. We used to say Go you now, and Sit you down, for example. Many frequently used expressions retain archaic and even obsolete forms, and "believe you me" is one such.
Yes. It would not have been, but having been in use for almost a century, it is now. You see, it became a phrase used for emphasis, and as such, doesn't have to follow the exact rules.
That said, many do find it annoying, and it generally connotes a lower educational level of the user, who is blissfully unaware that there could be any controversy about it.
Yes, I believe so.
"That was wrong" is a grammatically correct sentence.
This sentence is not grammatically correct. For the sentence to be grammatically correct, the space between "in" and "to" would have to be removed. Therefore the sentence should read "They are into skating."
The sentence is grammatically correct.
Yes, it is grammatically correct.
'Rhoda's crazy' is grammatically correct.
'What a drunkard you are' is a grammatically correct English sentence.
That sentence is grammatically correct.
This sentence is grammatically correct.
Yes, this sentence is grammatically correct, but you really don't need the second "that."
No, the sentence should be "Do you have any pens?" to be grammatically correct. "Pen" should be pluralized in this case.
This sentence is grammatically correct.