About the syntactical function and meaning of "up" in "man up" or "lawyer up"

Solution 1:

Am I correct in my thinking that up is an adverb here meaning into the desired or a proper condition, as quoted earlier?

There is more than one name for it (see the last paragraph in this answer). Yes, some call them adverbs. This book, on the other hand, calls them "adverbial particles". I use the term "particle". It doesn't make sense to try to define a particle (instead of defining the entire phrasal verb), because particles are:

a word or a part of a word that has a grammatical purpose but often has little or no meaning
_{Cambridge Dictionary}

So the OED defines "man up" and "lawyer up" as phrasal verbs. This is how it's done in all dictionaries. OED has two definitions for "man up":

• trans. To supply with the full number of workers required.
• intr. colloq. To demonstrate manliness, toughness, or courage when faced with a difficult situation. Also: to take responsibility; to own up. Frequently in imper.

As for "lawyer up", OED defines it as:

• intr. orig. U.S. Police slang. to lawyer up: to request a lawyer when being questioned by the police, often implying a probable lack of cooperation with the investigation; (also more generally) to hire a lawyer.

Is this linguistic pattern of forming new phrasal verbs an ascending trend in American English that is likely to be accepted into formal English any time soon?

A trend in American English? While you're not the first to think that (this source calls them "American parasites"), phrasal verbs are not just an American thing:

We can look at evidence about particular phrasal verbs in the OED, and find that some of these were indeed first recorded in American English, including check up (on), meet up (with) and win out. Others were in the language long before Americans could be blamed for them: stand up for is first recorded in 1608, in King Lear; help out in 1600. Others are relatively new, but with no evidence of being particularly American in origin or use: all the quotations for man up in the OED are from twentieth-century British texts, while speed up is first recorded in 1894 in the Westminster Gazette, with later quotations from a mixture of British and American texts.
_{Phrasal verbs: ‘a process of the common, relatively uneducated, mind’?}

While some style guides hate phrasal verbs, this doesn't mean that they aren't used in academic writing anyways. Look at the examples here, and then search Google Scholar.

Solution 2:

1. That appears to be the closest definition. Although, the examples given are somewhat informal.

2. I believe it would be considered an ascending trend, but I don't think it is likely to be accepted into formal English anytime soon. Although, it's widely understood here in America.

"Man up" means be more manlike, whatever that means. It's often used in the context of taking responsibility, being tougher, having more courage.

"Lawyer up" means getting a lawyer to represent you, as opposed to becoming more like a lawyer, or learning to be a lawyer.

Another similar phrase used in America is "suit-up" which means get prepared or literally get your suit on.

So the definition you chose (#5 Into the desired or proper condition) appears to be the best definition that fits these examples.