In the many examples found by Reverso in live bilingual texts online at http://context.reverso.net/translation/italian-english/stai+buono, stai buono is 'be quiet' and stai fermo is 'stay/stand/be still'. Don't trust Duo on this one. Check out the link.
This is what you would say to calm down a child for a moment, - e.g. to be able to tie his/her shoelaces. As "buono" means good / well / nice / fine the literal meaning would be something like:
Stay nice for as moment.
But what would be the closest translation if we think about how this would normally be phrased in English? Depending on the situation, probably rather something like:
Stand still . . / Behave yourself . . or Be quiet for a moment.
Formal You: "Stia (Lei ) buono per un momento" but it seems to me a little odd .. the sentence itself is informal (it's ok if it directed to a father in law)... would be better "Si calmi per un momento". P.S Stà is a contraction for Stai (imperative second person) Sta (egli sta) is the third person but of the indicative present
Yes is an irregular verb: http://www.zanichellibenvenuti.it/wordpress/?p=911 for the conjugation take a look: http://www.italian-verbs.com/verbi-italiani/coniugazione.php?id=11078 for the use of sta or stà : http://www.treccani.it/enciclopedia/sta-o-sta-sta_(La_grammatica_italiana)/
No. Duo wrote stai, not sta nor stia (see @Francesco), so it can only be the 2nd person singular. See http://www.wordreference.com/conj/ITverbs.aspx?v=stare.
In those conjugation tables note the 3rd person imperative and subjunctive. In every verb they are identical, because one does not give orders in formal speech so the subjunctive is used - in archaic English stia buono is something like "one may be quiet", and in modern Italian it sounds bizarre because stai buono is typically said to children.