AP style continues to use health care, two words. We differ from the dictionary; you can choose whether to use AP style or the dictionary's preference.

Just capitalize it. 

You're correct that the introductory clause doesn't agree with the subject, so the sentence is problematic as written. Adding "for everything" solves the grammatical part of the problem. But you still have two issues. One: "For everything" is an overly broad term (often used, but that doesn't make it a good thing). Does Acme Hospital handle obscure cancers, for example? If not, "for everything" is wrong. Two: The clause is overly long and hard to wade through, and it delays getting to the main action. I'd recommend rephrasing it. Something like:

Robotic surgeries. A state-of-the-art emergency room. Comfortable private rooms and maternity suites. Those are just a few of the reasons more people are choosing Acme Hospital.

or

For so many reasons, more people are choosing Acme Hospital: robotic surgeries, a state-of-the-art emergency room, comfortable private rooms and maternity suites, and more.

We typically use a link shortener to avoid these problems. 

From our official dictionary (Webster's New World College Dictionary):

pizzeria  n. [[It < pizza, pizza + -eria, -ery]] a place where pizzas are prepared and sold 

pizzaria  n. disputed sp. of pizzeria 

No, those are both lowercase in those uses. 

This is correct: Use no period at the end of a subhead, but do use a period at the end of each item in a bulleted list. 

You also surmise correctly that question marks are treated differently. We use a question mark (when it's the appropriate punctuation, of course!) at the end of both subheads and bulleted lists. However, if you use one question mark at the end of an item in a bulleted list, then use the same structure for each item and end each with a question mark.

We are discussing:

Yes, although typically AP doesn't use the formal name of a channel. We'd say the AP Stylebook's YouTube channel.

Second reference would be de la Vergne said. But if it leads a sentence, De would be capitalized.

For any ranking, use numerals. So it's top 5.

We'd probably try to avoid using it twice. We might write something like over two months in 2018, from May to June.

Yes, if you're certain that the reader will know that you're talking about the same group. But we would not capitalize veterans.

A Wikipedia entry lists 68 different things that can be abbreviated PD, ranging from People's Democracy, the weekly newspaper of the Communist Party of India, to peu difficile,  a French climbing/mountaineering grade. Parkinson's disease is among them, but the AP would never use it in stories for a general readership. For a specialized readership -- one that would know what you're talking about reflexively -- there's no harm in doing so. 

We use double quotes.

In order of your questions:
1. The plural of deer is deer -- no `s.' So the apostrophe is placed before the s.
2. Whole Foods, to quote the Stylebook, is a noun "plural in form, singular in meaning,"  like the United States. So: Whole Foods' policies.

Your second example is correct, but your first is wrong. The entry says "all references to the U.S. system" should be capitalized, which includes Social Security numbers.

The closest we get to an edict on such matters is this: "Avoid awkward placements of the time element, particularly those that suggest the day of the week is the object of a transitive verb: The police jailed Tuesday. Potential remedies include the use of the word on, rephrasing the sentence, or placing the time element in a different sentence." 

Regardless, your first variation (and especially the second version) is fine; the second seems awkward, as if the date was shoe-horned into the sentence; and the third, though it would appear to violate the AP's guidance, is clear and does not seem objectionable.

For our purposes, units of measure include those that describe physical dimensions (tablespoons, miles, etc.), not measurements of time.

Save off is redundant. We'd  go with on, which points to what is on sale.

Highest-quality is a compound modifier preceding a noun, so we would hyphenate.

Arguments can be made for either but we generally leave out the apostrophe because it doesn’t alter the meaning of what’s there or add to the phrase. 

We would go with Russian native, just as we would go with German native instead of Germany native.

Without commenting on whether FIP is a worthy acronym, we have to rule that your client is right. The Stylebook says the acronym for very important persons is VIPs. So FIPs, it is.

It's fast charging stations.

If you took this sentence and deleted the word both, there would be no need to use both an and a -- "exciting and stressful" would serve as a kind of compound modifier. But both emphasizes the contrast between exciting and stressful, so we'd use an own indefinite article for each one: "…can be both an exciting and a stressful time"