"You busy traveler, you" - what is that called
Keep those shoes on. You busy traveler, you.
Let's take each of your points one by one.
(1) "Those shoes" versus "your shoes"
Both are grammatically correct. "Those" shoes changes the tone. It is not true that using "those" instead of "your" always makes the tone more lighthearted, but in this example that's exactly what it does; it makes the tone more playful. It does this by making it less accusatory. Saying "your shoes" implies that "you" are responsible (and thus at fault if you fail to comply). "Those" redirects the blame elsewhere and thus makes it less likely to put the reader on the defensive.
(2) Period after "...shoes on."
You are correct; the period is not necessary. "Keep those shoes on." is a complete sentence, but "You busy traveler, you" is not a complete sentence. By omitting the period after "...shoes on", you now have one longer sentence: "Keep those shoes on, you busy traveler, you." With that being said, on a large sign/poster like this, grammatical correctness is not necessarily of greatest significance. The author likely chose to use a period for emphasis, to enforce a larger pause, or even just because it looked better visually. While this wouldn't be appropriate in formal writing, greater creative license is granted to signs and other short form writing like this.
(3) "You busy traveler, you."
Adding the "...you" at the end definitely increases the playfulness. It's meant to be jovial and playful.
It is used as vocative according to OED.
As vocative, chiefly in apposition to a following noun or noun phrase. Also in contexts expressing reproach of or contempt for the person addressed often emphasized by being placed or repeated after the noun. Cf. thou pron. 2.
Here are some early examples from OED:
Fy, fy, you counterfait, you puppet, you.
1600 Shakespeare Midsummer Night's Dream iii. ii. 289
You asse you, d'ee call my Lord horse?
1606 G. Chapman Gentleman Vsher iii. sig. D4v
ploce in rhetoric is applicable for this type of emphasis, but is uncommon.
A rhetorical term for the repetition of a word or name, often with a different sense, after the intervention of one or more other words.
diacope is even better for this type of emphasis.
Diacope is when a writer repeats a word or phrase with one or more words in between. A common and persistent example of diacope is Hamlet’s:
To be, or not to be!
Another famous example:
Free at last! Free at last! Thank God, Almighty, we are free at last!
— Martin Luther King, Jr.
The repetition of you in such "exclamatory" usages isn't particularly common. It took me a while to find these written instances in Google Books...
"Why, you idiot, you," cried Borton, jumping to his feet. "What can you know of my feelings?"
"I just wanted to tell you I love you, you wonderful person, you!
It's an "informal" usage. Probably intended to convey lightheartedness in OP's context.