This is actually something I could write intelligently about! However it's midnight and u/Trevor_Culley already has, here:

https://www.reddit.com/r/AskHistorians/comments/fkf5gg/why_did_the_babylonian_empire_deport_defeated/?utm_medium=android_app&utm_source=share

I believe this addresses a lot of your questions, though maybe not all exactly. In general, deporting large numbers of people after major sieges and campaigns was standard operating procedure throughout near eastern history. The neo-Assyrian empire, who preceded the neo-Babylonian empire, made it an art form, deporting an estimated 4.5 million people around the Assyrian empire over 300 years (see: History of the Ancient Near East by Marc van de Mieroop. If this subject interests you this is a good starting point, or a Very Short Introduction to the Ancient Near East). Incidentally this is a major driver of the Aramaic language and script spreading and becoming the lingua franca. These people were an important source of labor and population growth. They were provided for on the journey, as administrative documents address, and on arrival were not generally slaves. They worked for pay, and they and their descendants could obtain relatively high office. Local communities and families were kept intact. Karen Radner in various places (see: A Very Short Introduction to the Assyrian Empire) describes the process, at least for Assyria, as not so much solely punitive but as a pragmatic reallocation of resources and a practical way to quell rebellious attitudes. Would the deported populations agree? Psalm 137, especially the bit at the end they omit in church sometimes, demonstrates they didn't love it.

That is, notably, Babylon's doing, not Assyria. I can't speak so much on the differences in deportation policies, but Babylon could actually be harsher to rebellious kingdoms like Judah. The Assyrian heartland was relatively poor in terms of agricultural production so they couldn't afford to be too destructive to conquered kingdoms because they needed the resources. The Babylonians could, though.

I've talked about the idea in general but the linked answer talks about the specifics of Judah.

As an aside, under Nebuchadnezzar Babylon was a massive, cosmopolitan world capital, possibly over 200,000 people (and possibly the largest population of a city until Rome during the roman empire). Not all the deportees ended up in the city proper, as you'll read, and 20,000 people would still be disruptive, but it definitely could handle a pretty fair influx.

Like u/OldPersonName suggested, I think my other answer addresses most of these, but since I'm here, I'll address some of the other specifics that you asked about.

How the heck does an invading foreign army force the deportation of (possibly) 20,000 people?

In the least snarky way possible: If you have 10,000 soldiers with armor and spears and 20,000 disarmed people, including elderly and children, the soldiers are gonna win that one every time. That said, as I kind of indicated in the linked answer, it wasn't 20,000 all at once - and it wasn't 20,000 all to Babylon. One event that I didn't cover in that other thread is the final rebellion against Babylon after the destruction of Jerusalem.

2 Kings and Jeremiah describe a final revolt against Gedeliah, the local governor appointed by the Babylonians, around 582 BCE. Judeans who had taken refuge in the countryside or even fled Judean territory altogether returned to start working the land, but a small cabal of resistors tried to organize another revolt and assassinated Gedeliah. The Bible doesn't record another Babylonian deportation, but it does describe how many of the remaining Jews, including the prophet Jeremiah, fled to Egypt. These refugees were settled far to the south, on the island of Elephantine, as a military colony in service to the Pharaohs (and later to the Persians).

So there's at least three big movements out of Judea in the early 6th Century, and only two (admittedly the largest) of them went to Babylon.

20,000 Jewish people suddenly arrive, do the Babylonians just... absorb that into the population no problem, or were there conflicts like sometimes happens in modern times with countries absorbing a lot of refugees.

As OldPersonName said, Babylon itself was an order of magnitude larger than the whole Judean exile population, but they weren't just settled in the city of Babylon and its environs. They were settled in Babylonia. We know from local archives in other cities that Jews were present in Borsippa, Ur, Uruk, Nippur, and other Babylonian cities.

How long did they intend to keep them? Forever? I know the exile ended when Cyrus of Persia conquered Babylon, but how could they have prevented people from simply leaving and making the journey back?

Well, it depends on what you mean. In all likelihood, the Babylonians probably didn't expect to still have "Judeans" or "Jews" in Babylon 150 years later. Not because they would be released, but because they would be assimilated. Despite the apparent policy shifts of Cyrus' time, there's not much evidence to suggest that other deported groups returned home. In fact, of the potential millions shifted around in the Assyrian period, no obvious diaspora communities emerged. Interestingly, the Persians practiced deportation as well, but much like the Jews before them, many Greek deportees formed tight enclaves in Persian territory and maintained a variant of Greek culture and language for centuries. However, there's no historical references to Libyans, Thracians, Carians, or Lycians who suffered similar fates in the same campaigns forming diasporas.

The wiki article said only 25% of the population had been deported (at its highest estimated number of 20,000). "Only" implies that it could have gone higher without being remarkable!!

I would hazard a guess that the wiki article is phrased this way in contrast to a common religious interpretation of the Biblical narrative, not historical reality. Many religious believers have interpreted the Babylonian Captivity as the expulsion of all of the Judeans. That said, both the Assyrians and the Persians were documented deporting the whole populations of cities larger than Jerusalem, but rarely the same apparent level of desolating the countryside.

which group was the ... I suppose "driver" of that development: the people who were taken, or the people who remained? I'm assuming the people who remained, since the wiki article says this was the beginning of a permanent Jewish diaspora, but if there's a "king in exile" story of someone later returning to Judah and reestablishing the culture from the perspective of the exiled, I'd be interested in knowing that story.

In fact, you've got it entirely backwards. Judaism, or at the very least "more recognizable Second Temple Judaism" sprouted almost entirely from the returning exiles. The monotheistic followers of the God YHWH who remained in and around Judea formed the basis for the Samaritan community. After the exiles came back, this community was shunned for actually being entirely composed of foreigners who had started praying to YHWH after arriving. In reality, it was almost certainly some combination of both, but doubtlessly resting on a foundation of Judean/Israelite remainders.

The final composition and strict interpretation of the Torah, Jewish bloodlines and genealogy, and the construction of a new Jewish Temple were all the products of reformers who returned from Babylon (or at least the descendants of exiles). Most famously, this includes the legislator Ezra and the Persian-appointed governor Nehemiah.

There actually is a bit of a "king in exile" story, but it has very little to do with this. In the Book of Ezekiel, the prophet and/or his scribe meticulously dated many of his prophecies according to the Jewish calendar and the regnal years of King Jeconiah, who had been deported to Babylon after the first siege of Jerusalem in 598. Jeconiah's presence as a royal hostage is confirmed by a Babylonian document recording rations supplied to his household. Presumably, Ezekiel was not the only exile to track the years this way, though his work is the only example that survives. Jeconiah's grandson, Zerubabbel, was actually one of the first governors of Judea appointed by the Persians, but there's no evidence of the royal line continuing to rule after Zerubabbel, and even in his time it's not clear how much power he actually had.

Why would people who did leave Babylon at the end of the exile not return to Judah? The wiki article said that leaving was a "trickle" and many people who did return settled in what is now Israel/Lebanon/Syria area. After traveling thousands of miles, why not go back to your country of origin (or your parents/grandparents?) why get so close and then move in next door?

Well, imagine somebody whose grandparents moved to the United States from Poland because of persecution. They now have a steady business, but our hypothetical somebody takes advantage of a government program sending people to their family's home country. Through that process, they find a job in Austria and end up there instead. This is obviously a contrived fictional comparison, but I think it works well enough. By the time the return was really up and running, it had been about 80 years since the first large wave of exiles arrived in Babylonia. Most of the people able to return had no personal connection to Judea.

The presence of Jews settling in other regions west of Babylon does not necessarily indicate that they were heading back to Judea and got distracted either. The Persian Empire provided an immense logistical network to enable trade and travel, and people from the imperial core (ie Babylonia and the Persian home province) were able to find opportunities in the new regime. Likewise, new Jewish populations blossomed or expanded in eastern Persian territory like Media and Elam.

As for the return being a "trickle" that's fairly accurate, though you could also see it as a bit of a mirror image to the original deportation. At first, just after Cyrus the Great's proclamation, only a small number of dedicated people returned, but many of these people came from an upper religious class and found themselves at odds with the descendants of people deported to Judea by the Bayblonians and other local groups. Despite claiming Persian approval to return and rebuild Jerusalem, these groups opposed the returnees and complained to the Persian administration.

Under Darius the Great, about 20 years later, the Persian government finally cleared up the paperwork issues and gave the Jews a royal writ to rebuild Jerusalem, and funds to do it. This brought another larger wave of return along with more Jewish officials appointed by the Persian government to oversee the project.

Was there something happening that made Judah undesirable to settle in?

Yep, it was desolate, undefended, and surrounded by hostile forces. Many small time local powers that had become entrenched in the Babylonian Period didn't want Jerusalem rebuilt because it would become their competitor. The Samaritans went even further and opposed some of the derivations in Jewish religion from their own and the desire of the returnees to either segregate or convert the Samaritans.

To that end, they kept complaining to the Persian government until Artaxerxes I ordered construction in Jerusalem to halt once again. This halt didn't last for too long, and either Artaxerxes I or his grandson, Artaxerxes II, oversaw another wave of returnees with Ezra and Nehemiah as they put the finishing touches (ie the walls) on Jerusalem.