Pages 100-110.

We've temporarily removed the headlines entry because it needs to be updated. In short, AP style capitalizes the first word and proper names/proper nouns and lowercases everything else. 

Long-haul truck driver.

Do you need to use the phrasing at all? Seems to just add extra words. And I'm not sure what the ellipsis is doing in the middle of it. Is this a transcription?

If you have to use it:  This reveals tensions between, on the one hand, many liberals [...] and, on the other hand, conservatives [...]"

That's what is called a quasi possessive. The apostrophe is used with a measurement followed by a noun (a quantity of whatever the noun is). Here's the section of the possessives entry:

QUASI POSSESSIVES: Follow the rules above in composing the possessive form of words that occur in such phrases as a day's pay, two weeks' vacation, three days' work.

Frequently, however, a hyphenated form is clearer: a two-week vacation, a three-day job.

Rephrase it as you suggest: 

"A Star Is Born" actor Bradley Cooper displays a...

It's correct as you have it.

It's optional. Use the comma or don't; both are correct. Using the comma puts a pause in the sentence flow and thus more emphasis on the first word. It also just allows readers to catch their breath. Your first sentence is very long. Giving a little pause at the start of the next sentence isn't a bad thing.

Someone certainly can be a party to an agreement. Don't say the person is a signatory unless the person actually has signed something related to the agreement.

The National Association of the Deaf says:

Hearing-impaired – This term is no longer accepted by most in the community but was at one time preferred, largely because it was viewed as politically correct.  To declare oneself or another person as deaf or blind, for example, was considered somewhat bold, rude, or impolite.  At that time, it was thought better to use the word “impaired” along with “visually,” “hearing,” “mobility,” and so on.  “Hearing-impaired” was a well-meaning term that is not accepted or used by many deaf and hard of hearing people.

For more: https://www.nad.org/resources/american-sign-language/community-and-culture-frequently-asked-questions/

The American Foundation for the Blind, the  Center for the Blind and Visually Impaired, and other groups do use the term visually impaired when relevant.

Purists have some complex, head-hurting arguments for when due to is OK and when it isn't. Those arguments are related to nouns vs. adjectives, don't have anything to do with trains, boats or planes, and don't seem worth getting into here. It's safe to say that for most purposes, due to is OK in place of because unless you are writing for an audience of very traditional grammarians. Still, I'd say that because is preferable simply because it sounds less stilted.

You definitely need the commas, since it's a nonessential phrase. Here's the entry.

The quote marks aren't necessarily wrong, but they're also not necessary. Given that, I'd prefer the no-quote-marks approach for those little words. It's less disruptive for the reader, and it's clear without the punctuation.

We're with you. No to trainings. Yes to training sessions, etc.

No hyphen in any use. 

AP style doesn't use the accent marks. You're certainly free to use it if you prefer.

Yes, that looks good. 

Yes, it definitely applies to those as well.

An eight-week course, but an 8-week-old child (numerals with ages). Yes, it's confusing. Yes, we're hoping to add clarity. It will take some time.

That's correct if you have only one daughter. Use the commas.

If you have two daughters and are talking about just one of them at a time, then it's

Our daughter Charlotte continues her work and our daughter Emily moved to Switzerland.

In AP style, we never put an entity's abbreviation or acronym in parentheses following the full name. Our guidance is that if you want to use the shorthand later in the story, it should be familiar enough to readers that they can understand it. Otherwise, don't use the shorthand and instead just use a term such as the university, the group or the agency on later references. Here's the relevant section of the abbreviations and acronyms entry:

AVOID AWKWARD CONSTRUCTIONS: Do not follow an organization's full name with an abbreviation or acronym in parentheses or set off by dashes. If an abbreviation or acronym would not be clear on second reference without this arrangement, do not use it.

Names not commonly before the public should not be reduced to acronyms solely to save a few words.

We're working on developing some consistent guidelines. It's a work in progress. I'd say Class C felony for now. (But does it do the readers a service to use that level of detail/terminology? How about just calling it a felony and then spelling out what it entails.)

Use the plural, hookahs.

Lowercase it.

It's hard to read that way. I didn't understand the meaning the first time or maybe the first two times I read it. I'd either add a hyphen or rephrase:

They make millions income-tax-free.
They make millions free of income taxes.

(And tips on how to do that would be appreciated!)