Has the Buddha taught anything about how one should approach fixing society?
Does the Buddha advocate for striving to make society better?
Does the Buddha have any practices for groups and societies that would lead society to less suffering?
Does the Buddha have any teachings about Justice and Social Justice?
Yes. The best is the sutta on World Turning Monarch (DN 26). In addition other two important sutta are Sigalovada Sutta (DN 31) and Parabhava Sutta (Snp 1.6). There are many other suttas on the values of a great country.
Good householder,
Depending and enjoying certain society, one has of course duties in ones relations of which are of many kind for a householder. One not coming after his duties increases debt and endangers to be cut off while still full of desires.
In regard of getting understood the means, ways and kind of approaches, judgement from a wise perspective, this generous work makes many things clear: Wisdom over Justice.
Particularly, if one part-takes in the Buddhas society, nourishes on it, even benefits from it, one would have those duties there:
"Now I will tell you the layman's duty. Following it a lay-disciple would be virtuous; for it is not possible for one occupied with the household life to realize the complete bhikkhu practice (dhamma).
"He should not kill a living being, nor cause it to be killed, nor should he incite another to kill. Do not injure any being, either strong or weak, in the world.
"A disciple should avoid taking anything from anywhere knowing it (to belong to another). He should not steal nor incite another to steal. He should completely avoid theft.
"A wise man should avoid unchastity as (he would avoid falling into) a pit of glowing charcoal. If unable to lead a celibate life, he should not go to another's wife.
"Having entered a royal court or a company of people he should not speak lies. He should not speak lies (himself) nor incite others to do so. He should completely avoid falsehood.
"A layman who has chosen to practice this Dhamma should not indulge in the drinking of intoxicants. He should not drink them nor encourage others to do so; realizing that it leads to madness. Through intoxication foolish people perform evil deeds and cause other heedless people to do likewise. He should avoid intoxication, this occasion for demerit, which stupefies the mind, and is the pleasure of foolish people.
Do not kill a living being;
do not take what is not given;
do not speak a lie;
do not drink intoxicants;
abstain from sexual intercourse;
do not eat food at night, at the wrong time;
do not wear flower-garlands nor use perfumes;
use the ground as a bed or sleep on a mat.
"This is called the eight-factored observance made known by the Awakened One who has reached the end of suffering.
"With a gladdened mind observe the observance day (uposatha), complete with its eight factors, on the fourteenth, fifteenth and eighth days of the (lunar) fortnight and also the special holiday of the half month. In the morning, with a pure heart and a joyful mind, a wise man, after observing the uposatha, should distribute suitable food and drink to the community of bhikkhus. He should support his mother and father as his duty and engage in lawful trading. A layman who carries this out diligently goes to the devas called "Self-radiant." [7]
[Note that this isn't given for stacks, exchange, other world-binding trades or entertainment that binds but for an escape from this kind of society]
Does the Buddha have any practices for groups and societies that would lead society to less suffering?
I do have an answer for this part, from Tibetan and Ch'an Buddhism, found in a Terma teaching passed on by the tertön Jamyang Khyentse Chökyi Lodrö. This teaching is about what happens-—and the suffering it causes-—during the Dharma-ending time (now).
Ananda pleaded with the Victorious One:
“Victorious One, because you consider things with great compassion, spare humankind this torment.”
And the Victorious One replied to the venerable Ananda: "I have this to say about it. While I was residing in the great expanse of space, I was thinking of and gazing on the beings of the world. So, listen! There is a teaching useful during the time when the dark age arises. If one writes of it, propagates it, and practices the "Great Responsiveness Meditation" and recitation, impurities and obscurations will be quickly purified. If one sincerely makes offerings to this spiritual text with flowers and incense, all sentient beings will benefit. By propagating this text and writing about it, one will obtain a good existence throughout all of one's lives."
The full English translation, from which the above quote is taken, was done by Stephen B. Aldridge (Ka-rma gSung-rab rgya-mtsho) with the assistance of Ngor Thar-brtse mKhan-po, bSod-nams rgya-mtsho Rin-po-che. It is available in full in my article “Jamyang Khyentse Chökyi Lodrö's Prophecy of Things to Come”
Great Responsiveness Meditation is the practice used by Avalokitasvara (referred to as “the One Who Hears the Cries of the World” in the following quote) to become enlightened, which Manjurshi asserts is the best practice for all of us today:
Great Assembly! Ānanda! Halt the puppet show Of your distorted hearing! Merely turn your hearing round To listen to your genuine true nature, Which is the destination of the Path that is supreme. This is the genuine way to break through to enlightenment.
It is the way that the innumerable Buddhas followed Straight to nirvana's gate. All Thus-Come Ones of eons past Succeeded by this method. Through this method, Bodhisattvas, Too, right now are gaining perfect understanding.
Among the people of the future, those who undertake A spiritual practice should rely upon this teaching. I myself became enlightened by this very method. He Who Hears the Cries is not the only one. The Buddha, the World-Honored One, made a request That I consider methods that will rescue beings Who in the Dharma's ending-time resolve their minds Upon attainment of transcendence and nirvana. The best of all the methods is the practice Taught by the One Who Hears the Cries of the World.
The sages who attained enlightenment by other means Were aided by the Buddha's awe-inspiring spiritual power, And each was specially taught how to abandon all affliction. Some of these paths are shallow, some go deep; these teachings vary.
I bow now in respect to all the Buddhas, and I bow To all their Dharma-treasuries and to the marvelous ones Who've put an end to outflows. And may beings of the future Be empowered so that they will have no doubts That this one method is the most accessible.
It is the easiest way to reach enlightenment. It is the teaching most appropriate For Ānanda and for the beings drowning In the Dharma's ending-time. They only need This practice of the faculty of hearing For them to break through to enlightenment, For it surpasses all the other methods. It is the genuine path to the true mind.(taken from: “The Śūraṅgama Sūtra,” Buddhist Text Translation Society, 2009, pgs 253-257)
Research "engaged Buddhism." It is a movement within Buddhism that was started by Thich Nhat Hahn and addresses this specific question.
