What is the purpose of the "a"s here?

The first and third “a” are “[‘a’-prefixing](https://ygdp.yale.edu/phenomena/a-prefixing),” where the “-ing” form of verbs is given an “a;” it is normally now attached with a hyphen. > Here we come a-wassailing, a-wassailing we go! > > *Edit (low-frequency word):* “Wassail” is a hot Christmas drink made of spiced cider, ale, or wine. “To wassail” is a verb for an old tradition where people went door-to-door offering wassail to their neighbors and singing. It is no longer used in most forms of English, outside of old songs or expressions. You can basically just ignore that it is there. It has little effect on meaning. The second is just a normal indefinite article. You couldn’t say, for example: > Brag is good dog. ❌ You need one in there.

In certain regions or dalects or times, these a sounds get added for rhythm, mostly, and emphasis. Very old English habit. The author is conveying this character’s way of speaking. A US Western character from the 19th century for example might say “I’m a gonna go chop some wood,” or “I see the stagecoach is a’ nearin’,” or “I’m rightly a’feared of that horse, I won’t go a ridin’ on him.” Some words have these as built in, in adjective form, for example aflame, ablaze, afraid, aghast, etc.

The second highlighted section is perfectly standard. “a” is just being used as an article. In the first example it’s an attempt to represent dialect.

I'm not sure what your source is for these, but they seem to be full of misprints, as you've posted similar examples of this before. Perhaps they were scanned in? That can frequently lead to missing or misinterpreted text. That is a similar mistake in the text. In this case, the original text used "Who's a-going to try." A minor difference, but the prefix "a-" in front of verbs is a fairly common, and mostly archaic, dialectical affectation. It's really just a speech mannerism Joe uses in these cases.

The “a prefix” is an old-timey thing popular in the American south and (mid)west and Appalachia. My grandfather spoke this way. Typically in text, the “a” is followed by a dash that connects it to the next word. “Oh, a-hunting we will go, a-hunting we will go…” “I ain’t a-fixin’ to do that right now.” It’s usually used in the sense of *preparing* or *about* to do something, but not always (as in your second example). It’s just an old colloquial mode of speech, basically.

For the second highlight: a is being used as a regular indefinite article, but the speaker left out the noun it's referring to in the second part of the sentence, making it sound a bit weird: Brag is a good dog, but Holfast is a better (dog).  A modern speaker would probably say something like: Brag is a good dog, but Holfast is a better one.

Sounds like "hillbilly" vernacular

It's similar to the conditionally silent t at the end of words in French; where if, and only if, the word ending in t is followed by a word starting with a vowel, the t is pronounced. I think this is a regional thing, but then so is the a-going.

They indicate how the characters are speaking.

The "a" might be a misprint, or it might be from a 16th Century English idiom. https://www.enotes.com/topics/great-expectations/questions/explain-brag-good-dog-but-holdfast-better-141897

This is Dickens developing the character of Joe. His readers would understand the social class and educational background of Joe through this way if speaking.

“It’s a me, Mario!”

it's-a me, mario

As a beginner in English, I’m curious about the possibility as follows: S1(the origin): Who is a guy that is going to try? S2: Who is a [going to try(=guy)]? It seems like the author attempted to avoid repeating ‘is’ and omit ‘guy that (is)’ in order to emphasize the subject’s mindset as a person. This is how I interpret that, but I’m not sure.

1. The first "a" is vernacular - a pattern of speech typical of a group with certain shared educational, cultural, socioeconomic, ethnic, or geographic characteristics. Without knowing the reference, I would guess that Joe (note use of first name) is of a lower status than Mr Jaggers (note the honorific, "Mr"). Mr Jaggers' tone is also interrogatory, suggesting he is in a dominant position compared to Joe. 2. In the second highlighted passage, it is clear that both Brag and Holdfast are dogs. Mr Jaggers omits the word dog because it is understood implicitly by the reader. 3. The manner of speaking and inappropriate use of "a" in the third highlighted section suggests the speaker is of a status similar to Joe; he is not well-educated, makes grammatical errors, and uses a stereotypical pattern of speech associated with his status. I know all of the above is correct because I recognise the novel. I do think that recognising such subtle clues from a written imitation of a spoken pattern of English would be quite difficult and would require a number of years to accomplish.