Is this "adoni" related to "adonai", the word used to replace the tetragrammaton?
Yes, one is literally "my lord", the other, "my lords", because God is plural in Hebrew, (but grammatically singular). Like אלהים, literally "Gods", but you say אלוהים רואה, not אלוהים רואים.
They have the same source, אדון, which can mean something like "sir" or "mister" today, but originally it meant someone revered, master.
That would sound incredibly stiff if actually uttered. אֲדוֹנִי, אָסוּר לָרוּץ כָּאן (‘running is prohibited here’) would sound way more natural. נָא is something you’d expect to see on a sign and definitely not addressed at a specific person.
This leason is on formal Hebrew- I would expect it to sound somewhat stiff/not the most natural
Yes, but this is just stiff enough to be almost exclusively written.
לא באמת. תמיד שומע ברחוב וגם אני משתמש ב"נא" כי זה נוח. במקום "בבקשה"