Your question is actually many questions, with some statements mixed in.
Is there anything the American people can do "right now" to help
ensure stricter rules of eligibility are put in place in addition to
this criteria for someone wanting to run for POTUS?
Not so much, no. There was some talk (very generally) about easing the Rules so the Governator might run for President, See article here. Arnold also lobbied in 2013 to run in 2016 but after his affair, he no longer had much in the way of allies. (Too bad, I like him much better than Trump, but I digress).
There's no reason why the 35 years of age couldn't be pushed to 40, or an age limit to prevent people in their 70s from becoming president, but writing laws like that takes some doing, like congressional approval, a bill approved in both houses, signed by the Prez or passed by 2/3rds majority. It takes some doing and wouldn't guarantee your result.
Seems the current criteria alone isn't really enough to ensure the
American democracy is not at risk for total corruption undermining the
entire system.
Perhaps laws could be written to be more strict about international business interests and people running for office. That might be closer to what your question is asking. We could call those theoretical laws "Trump" laws and that might be possible. Anyone with international interests can't run - I actually like that. That might disqualify Hillary and her Clinton foundation (though her Hubby is the one who started it) and it would disqualify Trump. That might be the closest to an answer to your question.
We only get the choice of whoever the RNC or DNC allow to run and that
strategy seems corrupted too in the sense of not being in favor of
American people's best interest and rather what their parties believe
is the best choice instead—we should be controlling that choice.
Lets back this up a bit. Trump considered running as an independent. The Republicans let him run as a Republican and he won. During the Primaries, he even hinted that he could run as an independent if they treated him unfairly.
And Bernie was an independent since the 1980s. He ran as a democrat, but he really wasn't a member of the DNC. That's part of why so many people liked him.
So you had a situation where an independent won the RNC and an independent came in a respectable 2nd for the DNC, and, not that long ago an independent (Ross Perot) challenged the 2 party system. It's not entirely true that we just have DNC and RNC. The race is somewhat open. Granted it's hard for an independent to win, but Trump arguably was exactly that and he won. Careful what you wish for, as so many have said.
it would seem to a typical voting American citizen such as myself to
wonder if the entire American democracy has a loophole allowing
foreign countries to manipulate our system.
There is a loophole. Free press and social media. Obama and the FBI knew that Russia had plans to influence the outcome of the election. Obama didn't address it publicly, perhaps he should have, but our nation is so divided along partisan lines, I'm not sure it would have done much good.
Look at it this way. The US has tried to influence who wins in other nations elections. What's fair for the goose is fair for the gander, so what Russia did, it's nothing that our nation hasn't done, but at the same time, Obama was right to reciprocate with sanctions. There has to be a response to that kind of act. It has to be taken seriously.
Now if Russia helps Trump win and Trump returns the favor to Russia, or if Trump had meetings with Russia offering stuff before the election, that should be (and in many cases is) illegal and an impeachable offense. Whether it works out that way in time, we'll have to wait and see. It's being investigated. Trump is denying accusations. Time will tell what comes of it.
You would think things such as fully disclosing all business tax and
financial records to the FBI and other US government entities and
going thru extensive searches at this level ensuring no conflict of
interest ever could occur by such a person would be standard practice
for someone wanting to run for election of commander in chief POTUS
for an entire nation.
I don't know all the ins and outs, but I'm pretty sure those laws already exist. It's covered under the Emoluments Clause. Trump had to sign over control of several of his businesses to his Sons and he's required to not share or learn too much information regarding those businesses.
It's been argued that Trump has pushed that clause farther than it should be pushed and going forward, congress can pass ammendments or laws to fortify that clause and make it harder for Someone like Trump to become Prez. I'm not predicting that they will but they could.