What are some examples of forgiveness, endurance, patience and restraint from Ramayana?
One story that comes to my mind is the story of Kushanaabha and his 100 daughters whose self-control maybe the reason why we are breathing vayu and are alive today.
There are two other stories from Vālmīki Rāmāyaṇa that illustrate forgiveness. One is narrated by Rāma to Sugrīva and the other by Sītā to Hanumān.
When there's dissent among everyone whether to offer refuge to Vibhīṣaṇa or not, Rāma narrates a story of how a bird offers itself as food to a hungry fowler by jumping into fire.
It is narrated how by a dove, its enemy (a fowler) when it came for a refuge, was received according to rules of hospitality and was invited for a feast with its own flesh. O, the excellent of monkeys! The aforesaid dove indeed accepted as a guest, the fowler as he came, although he killed its wife. How much more a man like me has to do? Hear the verses inculcating virtue so long ago, by Kandu, the son of a sage called Kanva, a great sage and a speaker of truth. O, Sugreeva, even an enemy, who formed a cup with his hollowed hands, a miserable person, a person who is begging and who is seeking a refuge should not be killed with the aim of not being cruel. An enemy who comes for protection against others, even if the aforesaid enemy is oppressed or arrogant, is to be protected by one who has a disciplined mind, even by abandoning one's life. If he does not protect rightly through his strength, by fear or by ignorance or by desire, it is a sin to be reproached by the world. If having not been protected, a refugee dies before the eyes of a man who is able to protect him, the former takes along all his moral merit and goes. In not protecting thus the persons who take refuge, there is a great blemish involved in it. It does not bestow heaven. It destroys reputation. It devastates strength and valor. I will follow the excellent words of Kandu. It becomes a very righteous thing, gives reputation, leads to heaven and the rewards appear consequently. He who seeks refuge in me just once, telling me that 'I am yours', I shall give him assurance of safety against all types of beings. This is my solemn pledge. O, Sugreeva, the chief of monkeys! Let him either be Vibhishana or even Ravana himself; I have given an assurance of safety to him. Bring him here.
At the end of the war, when Hanumān visits Sītā to relay the news of Rāma's victory, she tells him to spare the lives of Rākṣasi's who'd troubled her earlier, by citing the story of a tiger chasing a hunter and how a bear offers him shelter.
O Hanuma! There is an old maxim possessed of merit, actually uttered by a bear in the presence of a tiger. Hear it from me. 'A superior person does not take into account the sin of those who have committed an offence (against him). The vow of not returning evil for evil must be carried out at all costs; (for) the virtuous persons account good conduct as an ornament.'
Formerly a tiger ran in pursuit of a hunter. The latter climbed up a tree. There was bear already preached on a bough of the tree. Making to the foot of the tree, the tiger addressed the following words to the bear: "Look here, both of us are denizens of the forest. The hunter is our common enemy. Therefore, knock him down from the tree." The bear, however, replied: "Having reached my abode, the hunter has in a way sought asylum with me. I am therefore not going to hurl him down; I would be deviating from my duty if I do so." Saying so, the bear laid himself down to sleep. The tiger now turned to the hunter and said, "Push the bear down. I shall afford protection to you." The hunter thereupon pushed the sleeping bear. The bear, however, clutched at another bough and thus escaped from falling down. The tiger now addressed the following appeal to the bear: "The hunter sought to hurl you down and has thus wronged you. Therefore push him down." Though pressed by the tiger again and again, the bear refused to hurl him down and repeated the above quoted verse in support of his attitude.