We prefer not to use labels and instead use more words to spell out specifics. 

In our style, Bay St. Louis, following the guidance in this entry:

Abbreviate as St. in the names of saints, cities and other places: St. Jude; St. Paul, Minnesota; St. John's, Newfoundland; St. Lawrence Seaway.But see the entries for Saint John and Sault Ste. Marie.

D.C. and U.S. both have periods and no space between. 

In datelines Washington doesn't take D.C. Generally use District of Columbia within a story only for official designations, such as local government names, or to avoid confusion with other localities of that name. Washington should be used in most story references to the U.S. capital because of the name recognition globally. Use Washington, D.C., with the added abbreviation only if the city might be confused with the state. Do not use D.C. standing alone other than in quotations. On second reference, the district is acceptable. 

Yes, hyphenate in that use.

Here's the entry:

Do not use a hyphen between adverbs ending in -ly and adjectives they modify: an easily remembered rule, a badly damaged island, a fully informed voter.

Yes, that's just fine.

Yes, treat it that way.

It's one department, right? The department presents. The department is the subect and is singular, so it takes a singular verb.

Here's the entry. It's capitalized for both singular and plural. But: Treasury bills, Treasury yields.

Securities sold by the federal government to investors to fund its operations, cover the interest on U.S. government debt and pay off maturing securities. Because they carry the full backing of the government, Treasurys are viewed as the safest investment.

If you're asking whether every AP staffer gets it right every time, the answer is no. Nonetheless, ultrarich is our style.

Descriptive terms with no apostrophe, if used in a general sense. If you're referring to specific writers' desks or specific artists' lofts, use the plural possessive with an apostrophe. 

Sorry, but we don't have guidances for that.

Spell it out.

The line break is enough. No comma needed. I'm a little confused about what it means, though. It wouldn't be The Madrid Edition, Germany. Or is Spain the general topic and there are separate editions on Madrid, Barcelona, etc.? I'm sure it makes sense in the context of the overall project. 

Yes.

Because someone wasn't following style or wasn't aware of that point. Mistakes happen, even among AP staffers.

No. It's perfectly clear without hyphens.

There's not really a rule or a correct way. Is there a reason to capitalize word nerds? Was that the formal title of this informal group? I'd go with the first option, but lowercase unless there's a particular reason to capitalize it. Also, group takes a singular pronoun.

She saw the word "Hawaii" written in the sand.

We follow Webster's New World College dictionary on those words. I can't speak to the dictionary editors' thinking. Certainly if you prefer to use one word, you can do that. I'll note, though, that Merriam-Webster (entirely separate from Webster's New World College Dictionary) also prefers copy editor as two words. 

I wouldn't use it as a compound modifier. Hard to read. Don't make it hard on readers! Instead, your second option but without the hyphens: a market worth hundreds of millions RMB.

We don't use the word on its own, so we don't have a style for it. You can create your own.

We follow our own guidance as described in the pre- entry. Since preadmission isn't in Webster's New World College Dictionary (our primary dictionary), we go with pre-admission. If you prefer Merriam-Webster's preadmission, that's fine too. They're both right. Just different styles.

pre- 

Follow Webster’s New World College Dictionary. Hyphenate if not listed there.

Either is fine.

Lowercase in that use.