Your planet seems to be a little bit on the impossible side.
You want you planet to be habitable for humans, and yet you want it to experience really extreme temperature changes which might exterminate all natural life on the planet. Including the plants that take carbon dioxide out of the air and convert it into the oxygen in the air.
Of course humans with sufficiently advanced technology could live on your planet. But the techiques they would need to shield themselves from the temperature swings would insulate them from the planetary enviromental conditions so much that they might as well living in an artifical space habitat.
You want the length of the planet's year to be 120 Earth years.
The length of a planet's year, defined as its orbital period around the planet, will depend on various factors. The equation to calculate the length of a planet's orbital period includes the mass of the star, the mass of the planet (which will almost always be so small comapred ot that of fhe star it can be ignored), and the distance between them.
Stars have different luminosities. A star with a specific luminosity will have a specific distance where a planet will receive exactly as much light and heat from the star as Earth gets from the Sun, and so (other factors being equal) would have exactly the same average temperature as Earth. I call that orbital distance the Earth Equivalent Distance or EED.
And the circumstellar habitable zone around the star will extend inward toward the star and outward from the star around the EED. The circumstellar habitable zone is where a planet would receive enough radiation from the star for water to be liquid on the surface of the planet. If other factors are good, a planet in the habitable zone can be habitable for life.
So how wide is the circumstellar habitable zone around the Sun?
The various estimates seen here:
Show there is considerabledisagreement on wide the Sun's circumstellar habitable zone is.
To be safe, some writers might want to put their habitable exactly at the EED of their star to be certain that it is within the star's circumstellar habitable zone.
The average orbital distance of the Earth from the Sun is one Astronomical Unit (AU).
If a star has X times the luminosity of the Sun, its EED orbit will be at 1 AU times the square root of X.
If the star is 4 times as luminous as the Sun, its EED will be at 2 AU from the star.
If the star is 9 times as luminous as the Sun, its EED will be at 3 AU from the star.
If the star is 16 times as luminous as the Sun, its EED will be at 4 AU from the star.
If the star is 25 times as luminous as the Sun, its EED will be at 5 AU from the star.
If the star is 100 times as luminous as the Sun, its EED will be at 10AU from the star.
If the star is 1,000 times as luminous as the Sun, its EED will be at 31.6227 AU from the star.
If the star is 10,000 times as luminous as the Sun, its EED will be at 100 AU from the star.
If the star is 100,000 times as luminous as the Sun, its EED will be at 316.227 AU from the star.
If the star is 1,000,000 times as luminous as the Sun, its EED will be at 1,000 AU from the star.
The luminoisity of stars - at lest the main squence stars - depends on their mass. A difference in the mass of two stars of a specific percentage will cause a difference in their brightness which is significantly greater. The most massive stars are thousands of times as massive as the least massive stars, but are millions of times as luminous.
There are a few known planets which orbit their stars at distances of hundreds of AU, and which have years hundreds of thousands, or even about a million, Earth years long.
And there are stars with the right luminousity and mass that a planet with a year 120 Earth years long could orbit within their habitable zone.
So everything seems fine for the planet in your story.
It took billions of years for life on Earth to develop enough for planets to produce an oxygen rich atmosphere that multicelled animals, such as humans, could breath.
The Sun shone with a fairly steady brightness for those billions of years - otherwise it would have made Earth too hot or too cold and all life would have died.
For a planet to become habitable, it has to have existed for billions of years, and its star has to have shown with a fairly steady luminosity for all those billions of years.
Unless the writer wants to make the planet much younger than a naturlly habitable planet, which an advanced civilziatin has terraformed to become habitable with advanced technology.
And as it turns out, only some stars, main sequence stars similar to the Sun with a rather small range of mass and luminosity can shine with se teady luminosity for the billions of year necessary for a planet to naturally become habitable for humans.
Stephen H. Dole, in Habitable Planets for Man, 1964, discusses the types of stars suitable for having planets habitable for humans on pages 67 to 72.
And to paraphrase a gambler, "read it and weep". many science ficiiton writers and readers have found those "Dole full" limitations on the types of stars which could have naturally habitable planets very frustrating.
Take a spectral class F2V star, which is about at the uppermass limit according to Dole, and which some astrbiologists would consider to be too massive to have a habitable planet. According to the table at:
an F2V star would have a mass 1.46 the mass of the Sun and a luminosity 5.13 the luminosity of the Sun. With 5.13 the luminosity of the Sun, it would have an EED at a distance of about 2.265 AU. I found an online orbital period calculator.
Entering the mass of the stars as 5.13 suns, the mass of the planet as 1 Earth, and the semimajor axis as 2.265 AU, I get an orbital period of about 2.82066 Earth years.
It is possible that an Earth mass planet in the orbit of Mars might be habitable. The orbit of Mars has a semimajor axis of about 1.523 AU. So the Mars Equivalent Distance, or MED, of an F2V star should be 1.523 times 2.265 AU, or 3.449 AU. And that would give the planet an orbital period of 5.30016 Earth years.
Vega is a class A0Va star, with 2.135 the Sun's mass, and about 40 times the Sun's luminosity. So its EED and MED should be 6.324 times as far as the Sun's, at 6.324 AU and 9.6322 AU. The orbital period of a planet at Vega's EED would 10.8821 Earth years and at Vega's MED would be 20.4557 Earth years.
And Vega's calculated lifetime on the main sequence is only about a 10th that of the Sun's, and so roughly a billion years.
Component C of Beta Scorpii would be a class B star to try. It is spectral class B2V, 8 times the mass of the Sun, and 3,200 times the Sun's luminosity. It's EED would be 56.568 times that of the Sun, at 56.568 AU and its MED would be at 86.153 AU, with orbital periods of 150.396 and 282.674 Earth years.
The main star of Algol, Beta Persei Aai, has spectral class B8V, a mass of 3.17 Suns and a luminosity of 182 Suns. Its EED and Med would be 13.49 times a far as the Suns, at 13.49 and 20.546 AU, with orbital periods of 27.8236 and 52.2982 Earth years.
According to the table at:
A B5V class star would have a mass of 4.7 Suns and a luminosity of 589 Suns. So its EED would be 24.269 AU and its MED would be 36.962 AU, with orbital periods of 55.1385 and 103.636 Earth years.
A B4V class star would have a mass of 5.1 Suns and a luminosity of 776 Suns. So its EED would be 27.856 AU and its MED would be 42.425 AU, with orbital periods of 65.0907 and 122.341 Earth years.
A B3V class star would have a mass of 5.4 Suns and a luminosity of 977 Suns. So its EED would be 31.257 AU and its MED would be 47.823 AU, with orbital periods of 75.1882 and 142.293 Earth years.
A B2V class star would have a mass of 7.3 Suns and a luminosity of 2,692 Suns. So its EED would be 51.884 AU and its MED would be 79.02 AU, with orbital periods of 138.298 and 259.938 Earth years.
So presumably a B2.5V class star would be about right to have a planet with a period of 120 Earth years orbiting between its EED and its MED.
Class B stars stay on the main sequence for only about a few tens of millions of years.
So a writer who wants a habitable planet orbiting one of them could have an advanced civilization terraform the planet to make it habitable. It might take that civilization thousands of years, but that might be considered worthwhile if the planet remains habitable for millions of years. And possibly after the terraforming civilization abandons the planet a native lifeform might develope intelligence and civilization and possibly develop spaceflight in time to escape, or humans might decide to settle there despite there being only a hundred thousand years before disastrous stellar changes.
Or you could take a look at the Mohs Scale of science Fiction Hardness and decide that you will be content with your story having a scale of only 1, and not worry abut the astronomical plausibiity of your system.
Obviously if the planet has a year 120 Earth years long, the astronomical seasons will each be 30 years long. The meteorological seasons might be longer or shorter in various places on the planet. But such long seasons should cause each hemispehre to get colder and colder and colder during the winter, and hotter and hotter and hotter during the summer. It might be impossible for humans, or for any lifeforms at all, to survive such temperatures, even though they should vary much less than you stipulated in the question.
If the planet has a very eccentric orbit, that could cause seasons which were the same on both hemispheres of the planet at the same time. And the temperatures could get very cold during winter and very hot in the summer. But I doubt that anything could make them reach the extremes specified in the question.
And you might be interested in questions and answers about making seaons on a planet last much longer than its years.
How does a Game of Thrones-style hyperwinter occur?