By the end of DS9 the Cardassians were a defeated people, much more open to the Federation and the Federation were helping them get back on their feet. Even Damar before his death became a new man and even before Cardassias joining of the Federation the Cardassians were more open to peace talks with the Federation and more notably the Bajorans. Don't forget that the military dictatorship that ruled Cardassia was overthrown and the Detapo council (rulers of the people) took power.
The Ferengi if you don't recall ended up with Grand Nagus Rom and his son Nog is a Starfleet officer. Rom was going to change the Ferengi civilisation and make it more tolerable of womens rights and what not.
The Klingons have been steadily working their way to becoming Fed members every series we saw. They became Federation allies, fought by the Federations side in the Dominion war and best of all it's now led by Martok who's spent a lot of time with Starfleet and starfleet officers. His best friend and house member is Worf who served years in Starfleet, grew up on Earth and is now Federation ambassador to Quo'nos.
Romulus is destroyed, what was once a tightly controlled empire is now an Empire in turmoil looking for some kind of help and a new direction. If the Federation does well in aiding the Romulans there's every chance they will come to respect the Federation. Before Nemesis the Romulan senate was more open with the Alpha Quadrant, even the newly elected senate denied the use of Shinzon.
The main problem with the Cardassians and Romulans were the Obsidian order and Tal Shiar. Once the Obsidian order was gone we saw the Cardassians rising up and taking power and kicking out the military dictatorship. With Romulus destroyed and the Tal Shiar likely weakened or also obliterated the Romulans are more free to make their own decisions.
As for the Dominion, Odo has every opportunity to open the Founders up to being more tolerant of solids. He's got years of memories and emotions to share with the link including love for a solid. In time even if the Dominion does not join the Federation or disband itself and allow it's subjugated worlds to join the Federation it can at least have a peace accord with them. More likely though IMO the Founders may eventually agree to disband the Dominion in favour of the Federation.
I fully agree with many of the things you are saying here.
The Cardis appear to be on a clear road towards Federation membership, even though it might take 2 or 3 decades. The Romulans as you say are severly weakened and in desperate need of help, which I'm sure the Feds are more than happy to supply. As a result I also find it quite likely that the empire will become much more open towards the Federation, especially after the erradication of the Romulan establishment, which has been tradionally imperialistic and xenophobic. The Ferengi are a bit of a mixed bag. On the surface their core values cleary differ from Federation dogma, regardless of how progressive Rom's administration might be. (I also highly doubt he will stay in government for long, he will face severe opposition and he simply lacks the killer instinct to lead a galactic power.) However, in the end, the Ferengi are mostly interested in the bottom line, after all there could be huge economic benefits from being a member of the Federation - a massive, wealthy and industrious market. And many of the culutres in the Federation have large differences, just compare the Risans and Vulcans or the Bajoran's and the Zakdorn or the Betazoids and Benzites. The Ferengi could fit in if they give equal rights to women and adopt democracy.
The Klingons are another matter though. There might have been a lot of rows over power and ideology within the empire over the years but overall it has remained a stable and strong power. Right now they have little to gain from joining the Feds and unless something happens that severly changes their fortunes I'm sure they will just stay good and strong allies.
The Dominion is even more unlikely to ever join. For one it is a vast and ancient power that I just don't see submitting to anyone. They might become friendly towards the Feds with time, maybe even allies, if they indeed change into a more benevolent group but anything beyond that is a very long way off.
Outside of that I'm sure the Federation will continue to grow. With the development of new technologies, which, as many in this thread have pointed out seem to be just over the horizon, I'm sure distance will be less and less of a problem.
Ultimately though, they will have to deal with the Borg, a hostile superpower that will never change its ways and already controls like a fifth of the entire Milkyway. It seems unlikely that the results of Endgame had much of an impact on that and as we all know the Borg are very adaptable. Therefore, I doubt they stayed vunerable to Janeway's super tech for very long. If the Federation continues to grow and make allies at the rate it has been doing they will eventually flag up as a significant threat and lucrative target for the Borg. Even if that takes hundreds of years I can't see the Federation ever surviving an attack by 10 million cubes.
