How did Hermonie work exactly?

That's explained in the Deathly Hallows.

Magical-borns get the letter delivered by owl. Muggle borns with no magical family get the letter hand delivered by a Hogwarts faculty member, called the special messenger, who explains it all and demonstrates magic to the parents to prove it.

Snape explained that to Lily when they were kids and he was talking about how they would get accepted to Hogwarts when they are 11.

Lily: "And will it really come by owl?" Snape: "Normally. But you're Muggle-born, so someone from the school will have to come and explain to your parents."

Dumbledore was Tom Riddle's special messenger as well, to explain it all since he had no knowledge of his magical lineage and was in the orphanage.

We're never explicitly told who Hermione's messenger was, but the general consensus and other hints from outside of the books is that it was McGonagall.

It's not a huge stretch to assume that all muggleborns get a guide assigned who takes them on an outing to diagon alley. And they do have banks, obviously wizards have a way to exchange wizard money for human money so it has to work the other way around, too. Then again, it wouldn't have been hard to write that explicitly if that was indeed how it's supposed to work, and HP is the kind of world where they throw tons of insane shit at people and tell them to just deal with it.

And most likely, the author just didn't give it a single thought.

Pretty much yeah. The teacher would be sent to find the person and they would walk them through the steps. We've seen that in Harry's case, we see then that little blip into Dumbledore and Tom Riddle, and we see that in that video game. Though I don't know if you need financial assistance when her parents were dentists in England. Probably not poor. In a later book we even see them going to the bank I believe.

having buried her nose so far into the books that she effortlessly and immediately fixes Harry’s glasses upon meeting him

This happens only in the movie. In the book, she just mentions that she had "tried some simple spells", and that they had worked for her, but doesn't list what spells they were or demonstrate them.

Given that she came from ordinary humans, how does this work?

There is vague mention of "magical genes", though the exact heritability is not rigorously fleshed out. We can conclude that it isn't simply dominant or recessive, as both squibs and muggle-borns exist. It is generally surmised that muggle-born wizards must be the descendents of at least one squib, likely several generations back.

How did they know to go to Diagon Alley to get Hermonie’s wizarding supplies? How did they know how to find Diagon Alley?

For muggle-born students, a member of the Hogwarts staff hand-delivers the letter and explains the situation to both the student and their parents. A staff member would also accompany the family on their first trip to Diagon Alley, like Hagrid did for Harry.

Did they pay for her books? How did they pay for Hermonie’s supplies with British Pounds, since they obviously don’t have wizard money?

There is a short paragraph in Chamber of Secrets in which we see Hermione's parents exchanging Pounds for Galleons at Gringotts.

If there’s a financial aid program, why didn’t it help Ron?

The only time we see any sort of financial aid program at Hogwarts when Dumbledore visits the young Tom Riddle at the orphanage. Ron was poor, but not an orphan without financial support. Molly and Arthur could afford Ron's school supplies, just not brand-new or top-of-the-line ones. There's probably a deeper discussion to be had here on why "poverty" even exists in the magical world.

Why did they even humor the idea of a Wizard School wanting to teach Hermonie magic instead of writing it off as a joke or scam? How is it that Hermonie immediately started studying and completely skipped the “Holy shit, magic is real!” phase that should naturally come with all of this?

I feel like if Professor McGonagall shows up at your house and transfigures your table into a pig and back again, the evidence would be quite convincing :)

The staff member would likely also inquire as to what accidental magic the child has already exhibited. For many families, it is likely a relief to learn that their child's "strange powers" have an explanation, and that there are schools dedicated to refining and honing those powers.

For the muggle-borns themselves, they may have noticed that they were able to do things other children couldn't, but not have known that it was magic. So the revelation would be more of a "oh, so that's why I can do that" as opposed to "holy shit, magic is real".

A couple nitpicks:

Hermonie

Hermione

she is a witch despite coming from a No-Maj family

No-Maj is an american term. Since Hermione's family is British, they are muggles :)

Always viewed Hermione as J.K. Rowling's self-insert character.

That's really the only way that everything that she does makes any sense.

Answering those questions would make for an interesting spin-off, but as a writer I can tell you: we don't care about secondary characters at first, so she was just written, as Ron was, as a satellite for the protagonist, inspired by the know-it-all archetype. The most probable answer is that when the school enrolls a pure muggle-born, the same thing that happened with Harry Potter happened to Hermione : someone from the school showed up and accompanied her until she began her courses, and she probably read a lot in advance, starting from the very moment she got her wand and a grimoire. Now, If we want to make it interesting, we can speculate that her thirst for perfection comes from a deep insecurity and need to satisfy her parents' expectations. She probably must have been a miserable trophy child before magic delivered her, and yet it didn't, because she still caries this toxic perfectionism and façade. So, she's either bright, but under inherited, neurotic performative pressure, or she's just gifted and masking, which is more or less the same thing. And she can never say no, or admit that she is tired. Else, who would take an interest in her existence?