How did Ron open the Chamber of Secrets?
This is explained in the book, although not very well: Ron uses a heretofore-unknown talent for mimicry to exactly duplicate the sounds Harry made earlier in their adventure. From chapter 31, "The Battle of Hogwarts":
"But how did you get in [the Chamber of Secrets]?" [Harry] asked, staring from the [Basilisk] fangs to Ron. "You need to speak Parseltongue!"
"He did!" whispered Hermione. "Show him, Ron!"
Ron made a horrible strangled hissing noise.
"It's what you did to open the locket," he told Harry apologetically. "I had to have a few goes to get it right, but," he shrugged modestly, "we got there in the end."
Where Ron acquired these skills is not discussed, but it's clear that he's not actually speaking the language - he doesn't know what he's saying (although he assumes it's something like "Open"), he's just recreating the sounds.
Note that Ron's "strangled hissing" isn't translated into English, while every other occurrence of Parseltongue in the series has been; the fact that Harry never actually hears Parseltongue when it's spoken is a fairly major plot point in Chamber of Secrets.
This suggests to me that, unlike non-magical languages, there's a difference between Parseltongue's phonemes1 and the language itself; otherwise Harry would have unconsciously translated, and we would have read "'Open up,' Ron said" instead of "Ron made a horrible strangled hissing noise".
It would appear that the entrance to the chamber isn't coded to accept Parseltongue, per se, but rather to accept a spoken password that just happens to correspond to the phrase "Open up" when spoken in Parseltongue.
Wait, what? So you can speak Parseltongue without speaking Parseltongue?
Apparently. Although I agree that this is a bit inconsistent (why is this a hereditary trait if any idiot can learn to speak and understand it?), there is a certain amount of logic to it.
As far as I'm aware, the above quote is the only time Parseltongue is described - every other time we hear it, as previously noted, Harry automatically translates and we read English (or whatever language you're reading in). But Parseltongue gets a much broader treatment in the films, where we actually hear Harry speaking it on several occasions.
The films, or at least Goblet of Fire and Deathly Hallows: Part 1 actually bothered to hire Cambridge linguistics professor, Dr. Francis Nolan, to construct phrases of a real Parseltongue language2, with distinct phonemes3. If we assume that JKR's original Parseltongue also had distinct phonemes (and we have to, because otherwise this scene is impossible), then it's not inconceivable that Ron could have reproduced these phonemes by ear - this is how we learn to speak languages, after all.
So how does Dumbledore know it?
It's established canon that Dumbledore understands Parseltongue - he implies the ability in Half-Blood Prince when taking Harry into the Pensieve to visit the Gaunt family, and JKR confirmed it in a 2007 live-chat on Bloomsbury.com:
J.K. Rowling: Dumbledore understood Mermish, Gobbledegook and Parseltongue. The man was brilliant.
What's not made clear is how he's able to understand it. It could be that there's a magical component to his understanding - he duplicated the innate magical ability of Parselmouths, and he uses that to translate.
It's also possible that he learned how to interpret the language's phonemes. Although possible, this seems to me unlikely largely because of the resources required.
There are basically two ways Dumbledore could have reconstructed the language of Parseltongue. One of those ways is, essentially, code-breaking. But code-breaking when you don't have the key relies partially on complicated statistical techniques, but he'd be trying it out on a completely unfamiliar language - he may be able to determine the most commonly-used phoneme4, but he wouldn't know what that corresponds to in English.
The other way would be to stumble across a phonetic phrasebook, but the odds of such a book existing are slim. Given the already-small population of Parselmouths, how many of them do you think have the linguistic inclination necessary to create a phonetic translation of Parseltongue, let alone the desire to put in writing the secret language of Evil People. In a world where purebloods don't learn mathematics, I'm guessing the number of linguists is zero.
Even if Ron could have opened the Chamber, how did he get down there?
So, in Chamber of Secrets the tunnel to the Chamber is collapsed, and the way back in from the Girl's Lavatory is blocked by what Harry refers to as "a solid wall of broken rock." However, later in the book Harry returns that way with Ginny, and we learn that Ron has not been idle (emphasis mine):
After a few minutes' progress up the dark tunnel, a distant sound of slowly shifting rock reached Harry's ears.
'Ron!' Harry yelled, speeding up. 'Ginny's OK! I've got her!'
He heard Ron give a strangled cheer and they turned the next bend to see his eager face staring through the sizeable gap he had managed to make in the rock fall.
'But you're okay, Ginny,' said Ron, beaming at her. 'It's over not, it's - where did that bird come from?'
'He's Dumbledore's,' said Harry, squeezing through himself.
--Harry Potter and the Chamber of Secrets Chapter 17 "The Heir of Slytherin"
The fact that Harry's had to "squeeze" through indicates that the hole Ron made was not very large; it barely accommodated twelve-year-old Harry, described as scrawny and very underweight. It's unlikely that seventeen-year-old Ron and eighteen-year-old Hermione would have had any chance of getting through.
Even assuming the tunnel hadn't been cleared in the five years between this event and the events of Deathly Hallows, the other thing you need to remember is that Hermione and Ron are both much more accomplished at magic at age seventeen/eighteen than Ron alone was at age twelve. Considering all the feats we see her do over the course of the series, I have no doubt that Hermione alone would have been able to safely unblock the tunnel.
1 The basic sounds that make up a spoken language
2 The fact that Dr. Nolan doesn't list this accomplishment on his website shall not be remarked upon
3 There's also a dedicated fan project to expand on Nolan's work, called Stilio. It's a fascinating idea, although pretty dense if (like me) you're not a languages nerd
4 In English, that's /ə/, which in most dialects is a middle "a" sound, as in "comma"