Learn English – Is this usage correct? “Please get your parents to come tonight”

I'm sure you suspect the answer: The preferred phrasing is more subservient and less demanding (a role many waiters at fancy restaurants are encouraged to play) even though they mean the same thing.

Sir, I'm going to have to ask you to put out your cigar.

The previous sentence carries a few implications. One might expand it to:

Sir, I would never ask this if it was up to me, but those damn rules they make me follow FORCE ME to ask that you please put out your cigar. I don't blame you, of course.

That is, it's implied that the person asking isn't personally bothered by your cigar; he's just forced to ask you to stop by company policy.

In a less fancy resturant, one might say:

Sorry, but we don't allow smoking in here.

This is the same exact idea. "It's not up to me; it's company policy, sorry!"
It may seem silly (and it is, in some sense), but it probably does help the customer not feel like an idiot!

Yes, those are correct usages. Use viz. just as you would use namely. Wikipedia's examples:

• The main point of his speech, viz. that our attitude was in fact harmful, was not understood.
• "My grandfather had four sons that grew up, viz.: Thomas, John, Benjamin and Josiah."
• The noble gases, viz., helium, neon, argon, xenon, krypton, and radon, show a non-expected behaviour when exposed to this new element.

Viz. is short for the Latin videlicet, which means namely. Though I am a native English speaker, and I would not find this weird or inappropriate, it is uncommon, and most people would have to look it up in a dictionary. It would be better to use namely or that is to say instead.

Edit: I find this Ngram rather informative, too. Once, apparently, viz. was more popular than namely:

Most of the modern usages of viz. are not the abbreviation in question, but nearly all the older ones are.