Related question here
Worf has been to Minsk, he talks about it in TNG.
He also spent time in his parents' native Minsk, later recommending it to O'Brien as one of his favorite places on Earth.
Here is a partial answer:
From birth to the age of seven, Worf lived in various places in the Klingon Empire. After the Khitomer Massacre, he was rescued and adopted by Sergey Rozhenko.
IN TNG "Heart of Glory":
KORRIS: I do not wish to anger you. We are after all, brothers lost among infidels. Tell me, how is it that you come to this ship, that uniform?
WORF: Through an act of kindness. The Romulans attacked the Khitomer outpost. Everyone was killed. I was buried under the rubble and left for dead. A human Starfleet officer found me. He took me to his home on Gault and told his wife to raise me as his son.
KONMEL: How old were you?
WORF: Before the age of inclusion.
KONMEL: That young?
KORRIS: Gault is a farming colony.
WORF: When my foster brother and I were of age, we entered the Starfleet Academy. He hated it and returned to Gault. I stayed.
KONMEL: You have not spent much time among your own kind.
WORF: Hardly none.
in TNG "Sins of the Father":
KURN: That is the response of a Klingon. The response I would expect from my older brother. (collective gasps and a quick advert break)
KURN: I was barely a year old when you left for the Khitomer Outpost. You, our mother and father were not going to stay long. It was decided that I did not need to go. I was left to stay with our father's friend, Lorgh, until you returned. You never did.
WORF: The Starfleet officer that rescued me was told by the Klingon High Command that I had no living relatives.
KURN: They assumed that I was killed with the family at Khitomer. Lorgh had no sons. He took me into his family. It was not until I had reached the Age of Ascension that I was told the truth.
WORF: My father is accused of aiding and abetting the Romulan attack on the Khitomer outpost.
PICARD: The attack in which he himself was killed? But why now, after twenty years?
This shows that Worf is about 27 years old in the third season of TNG.
In DS9 "Let He Who Is Without Sin":
WORF: You know from the age of seven I was raised by humans on the farmworld of Gault? It was a sparsely populated world, no more than twenty thousand inhabitants. It often seemed as though everyone knew everyone else. But one thing was for certain, everyone knew me. I was the Klingon child, the uncontrollable one. I was the biggest, the strongest, most fearless child on the entire planet. I fought hard, played hard, I did as I pleased.
DAX: What changed?
WORF: When I was thirteen, I was captain of my school soccer team. We had made the championships, and I was determined to win. Near the end of the second half, with the score tied, my team got a corner kick. The ball sailed up high. Both I and one of my opponents, a human boy named Mikel, leaped up to head the ball. He had position, but I was determined to score. I remember laughing with excitement as I threw myself at him.
DAX: Go on.
WORF: The next thing I knew, the ball was sailing into their goal. I roared with triumph and turned around to Mikel to gloat, only to find him lying on the grass bleeding. Our heads had collided when we both went up for the ball. I had not feel the impact, but I had broken his neck, and he died the next day.
So this establishes that Worf lived on the planet Gault from the age of seven until at least the age of thirteen. He may have gone on visits to places off of Gault, possibly including Minsk on Earth, but Gault was his place of residence for those years.
It is possible that the Rozhenkos decided to move from Gault, possibly to Earth, soon after this incident, but Worf does not mention it.
Worf would have lived with the Rozhenkos wherever they lived until he entered Starfleet Academy. The usual age for entering Starfleet Academy should be about 17 to 22, so Worf would have lived with the Rozhenkos for about 4 to 9 years until he entered Starfleet Academy, presumably the campus in San Franciso on Earth, since no other campuses are mentioned in canon that I know of.
Worf would have lived at the Starfleet Academy campus while a cadet there. After being commissioned he lived on the starbases and ships he served on.
So the only time Worf might have permanently lived in Minsk would have been during the years before he entered Starfleet Academy, and only if the Rozhenkos moved there with him after he was 13.
In TNG "Family": Sergey and Helena Rozhenko beam aboard the Enterprise to visit Worf:
FEMALE [OC]: Enterprise, this is Earth Station Bobruisk. Two to transport aboard.
Bobruisk or Babruysk is a city in Belarus, about 80 miles from Minsk. So during TNG the Rozhenkos lived next door to Minsk by 24th century standards.
Thus Worf probably visited Minsk a few times while visiting his parents while in Starfleet. And he may have lived there for a few years before entering Starfleet Academy.
The enthusiastic way Worf recommended Minsk as a place to live in DS9 "What You Leave Behind" indicates he was familiar with Minsk instead of relying on second hand information.
It was clearly established (in the various EU novels) that Worf's adopted parents, after leaving the Earth colony on Gault, had retired to a small farmholding in Minsk. Worf traveled with them and spent some time there prior to joining Starfleet. Evidently he grew to like the city and its environs.
When Helena Rozhenko opened the door of her modest farmhouse in Minsk, she let out a girlish squeal of delight that did not remotely seem to match her exterior as she saw Worf standing in the doorway. She threw her arms around him before he was able to get a word out, and as courageous as the Klingon was, he had to admit he felt safer in the elderly woman's embrace. "Sergey!" she cried out, summoning her husband. "Worf, why didn't you tell us you were coming ... ?"
Zelik had been hearing stories about the Rozhenkos’ Klingon boy all of his life. The farming community on Gault was a very close-knit one. While the Rozhenkos had retired to Minsk on Earth by the time Leybenzon was a teenager, the stories about them lingered. Not all those stories were pleasant ones. Worf had accidentally caused the death of a fellow teenager during a soccer game.