Truly, and with all due respect to those who created it, there isn't much about that acronym that makes sense.

Since it is, indeed, commonly referred to and written as MOSAiC, I'd stick with that. But I'd capitalize Drifting in the full name. Compromises and all.

As you probably know, we don't enclose acronyms or abbreviations in parentheses after the full name. Generally, we say that if it doesn't make sense without the parenthetical reference, then don't use it at all. But again: compromises for the sake of the reader. In this case, I'd write something like this:

The Multidisciplinary Drifting Observatory for the Study of Arctic Climate, known as MOSAiC, was a yearlong expedition ...

As you have it. Here's the guidance: 

PLACEMENT WITH QUOTATION MARKS: Inside or outside, depending on the meaning:

I'd use earlier. Nine months earlier.

I'd use the headline rules for news articles. But you can do differently if you prefer, of course.

We don't use "for" constructions unless a person has been convicted. It implies guilt. Instead: He was arrested on a drunken driving charge (if charged) or He was arrested on suspicion of drunken driving.

Webster's New World College Dictionary doesn't recognize it, though it does appear to get some use elsewhere. Vandalism is much better.

Periodical most often refers to a publication schedule. Periodic is better in your example.

The sentence doesn't make sense to me, though. I think you're better off to delete "the exception of." Thus: He likes working at home with periodic commutes to the office.

I'd use lowercase for all, but we don't have a specific style.

I would use the prepositions. Among the reasons: It avoids the vision of poor Christmas being eaten at the dinner table.

Not all in one place. We cover the concepts generally in a number of separate entries, including disabilities, gender and sexuality, race-related coverage, among others.

The second. I wouldn't argue with the first, though. 

best-of-show nomination

Yes, as a title before a name.

I'd use quote marks in the first reference. No need in later references.

That's not a topic that we cover.

First, I'd paraphrase rather than using the clunky direct quote. You can convey the meaning more clearly and with fewer words and no dicey punctuation. One example:

Especially in light of the pandemic, he said, people nationwide are trying to determine the best use of office buildings.

If for some reason you need to use the direct quote, I'd delete the comma after asking and otherwise leave it as you have it. There's no perfect solution here, other than to paraphrase.

Chart-topper

I'd rewrite it. If the plural is an option, why not just use it: Inmates have plenty of options ...

We don't abbreviate it.

We are fine with hypens for ranges in some uses. But really, we'd prefer to write it out in your example. Also, a nonessential clause construction is better in this case, unless the spider also lays other eggs that don't hatch in seven to 11 days.

 The spider lays 45 to 85 eggs, which hatch in seven to 11 days.

... 6 feet ...

We wouldn't use the term, so we don't have a style. In what context would you use it? If you have to use it, I'd hyphenate.