Parashurama, Lord Hanuman, Jambhavan and Lord Shiva appears in both(Ramayana and Mahabharata) What has been mentioned about Lord Rama in the Mahabharata?
In the Vana Parva of Mahabharata there is Ramopakhyana section ie. Ramayana is told there by Markandeya to Yudhisthira. One example from that section is:
"Markandeya said, 'O bull of the Bharata race, even Rama suffered unparalleled misery, for the evil-minded Ravana, king of the Rakshasas, having recourse to deceit and overpowering the vulture Jatayu, forcibly carried away his wife Sita from his asylum in the woods. Indeed, Rama, with the help of Sugriva, brought her back, constructing a bridge across the sea, and consuming Lanka with his keen-edged arrows.'
"Yudhishthira said, 'In what race was Rama born and what was the measure of his might and prowess? Whose son also was Ravana and for what was it that he had any misunderstanding with Rama? It behoveth thee, O illustrious one, to tell me all this in detail; for I long to hear the story of Rama of great achievements!'
"Markandeya said, 'Listen, O prince of Bharata's race, to this old history exactly as it happened! I will tell thee all about the distress suffered by Rama together with his wife. There was a great king named Aja sprung from me race of Ikshwaku. He had a son named Dasaratha who was devoted to the study of the Vedas and was ever pure. And Dasaratha had four sons conversant with morality and profit known by the names, respectively, of Rama, Lakshmana, Satrughna, and the mighty Bharata. And Rama had for his mother Kausalya, and Bharata had for his mother Kaikeyi, while those scourge of their enemies Lakshmana and Satrughna were the sons of Sumitra.
Discussing about Ram was well explained by @Tezz and @Ram in their answers. Here, I found another example how Ramayana is older than Mahabharata.
कर्मणैव हि संसिद्धिमास्थिता जनकादय: |
लोकसंग्रहमेवापि सम्पश्यन्कर्तुमर्हसि || Shlok 20, Chapter 3, Bhagavat Geeta.
Lord Krishna had given an example of king Janaka for how to attain a perfection through karma-yoga in ShrimatBhagvatgeeta.
Sri Krishna mentions Sri Rama (himself in previous avatara) in Bhagavad Gita 10.31
Ramayana characters are mentioned in Mahabharata
Hanuman met Bhima because he was intoxicated with strength.
"Then knowing him (Bhima) to be intoxicated with strength, and proud of the might of his arms, Hanuman, slighting him at heart, said the following words, 'Relent thou, O sinless one. In consequence of age, I have no strength to get up. From pity for me, do thou go, moving aside my tail.' Being thus addressed by Hanuman, Bhima proud of the strength of his arms, took him for one wanting in energy and prowess, and thought within himself, 'Taking fast hold of the tail, will I send this monkey destitute of energy and prowess, to the region of Yama.' Thereat, with a smile he slightingly took hold of the tail with his left hand; but could not move that tail of the mighty monkey.
Hanuman later told about other Ramayana characters.
Thereupon Hanuman said, 'O represser of foes, even to the extent of thy curiosity to know me, shall I relate all at length. Listen, O son of Pandu! O lotus-eyed one, I was begotten by the wind-god that life of the world--upon the wife of Kesari. I am a monkey, by name Hanuman. All the mighty monkey-kings, and monkey-chiefs used to wait upon that son of the sun, Sugriva, and that son of Sakra, Vali. And, O represser of foes, a friendship subsisted between me and Sugriva, even as between the wind and fire. And for some cause, Sugriva, driven out by his brother, for a long time dwelt with me at the Hrisyamukh. And it came to pass that the mighty son of Dasaratha the heroic Rama, who is Vishnu's self in the shape of a human being, took his birth in this world. And in company with his queen and brother, taking his bow, that foremost of bowmen with the view of compassing his father's welfare, began to reside in the Dandaka forest. And from Janasthana, that mighty Rakshasa monarch, the wicked Ravana, carried away his (Rama's) queen by stratagem and force, deceiving, O sinless one, that foremost of men, through the agency of a Rakshasa, Maricha, who assumed the form of a deer marked with gem-like and golden spots."
Hanuman further told his story in another section.
Hanuman said, 'And after his wife was carried away, that descendant of Raghu, while searching with his brother for his queen, met, on the summit of that mountain, with Sugriva, chief of the monkeys. Then a friendship was contracted between him and the high-souled Raghava. And the latter, having slain Vali installed Sugriva in the kingdom. And having obtained the kingdom, Sugriva sent forth monkeys by hundreds and by thousands in search of Sita. And, O best of men, I too with innumerable monkeys set out towards the south in quest of Sita, O mighty-armed one. Then a mighty vulture Sampati by name, communicated the tidings that Sita was in the abode of Ravana. Thereupon with the object of securing success unto Rama, I all of a sudden bounded over the main, extending for a hundred yojanas. And, O chief of the Bharatas, having by my own prowess crossed the ocean, that abode of sharks and crocodiles, I saw in Ravana's residence, the daughter of king Janaka, Sita, like unto the daughter of a celestial.
And having interviewed that lady, Vaidehi, Rama's beloved, and burnt the whole of Lanka with its towers and ramparts and gates, and proclaimed my name there, I returned. Hearing everything from me the lotus-eyed Rama at once ascertained his course of action, and having for the passage of his army constructed a bridge across the deep, crossed it followed by myriads of monkeys. Then by prowess Rama slew those Rakshasas in battle, and also Ravana, the oppressor of the worlds together with his Rakshasa followers. And having slain the king of the Rakshasas, with his brother, and sons and kindred, he installed in the kingdom in Lanka the Rakshasa chief, Vibhishana, pious, and reverent, and kind to devoted dependants. Then Rama recovered his wife even like the lost Vaidic revelation.
Then Raghu's son, Rama, with his devoted wife, returned to his own city, Ayodhya, inaccessible to enemies; and that lord of men began to dwell there. Then that foremost of kings, Rama was established in the kingdom. Thereafter, I asked a boon of the lotus-eyed Rama, saying, 'O slayer of foes, Rama, may I live as long as the history of thy deeds remaineth extant on earth!" Thereupon he said, 'So be it. O represser of foes, O Bhima, through the grace of Sita also, here all excellent objects of entertainment are supplied to me, whoever abide at this place. Rama reigned for the thousand and ten hundred years. Then he ascended to his own abode.
In this chapter of the Shanti Parva of the Mahabharata, Vishnu's Aniruddha form (which I discuss here) tells Narada about the different incarnations he is going to take in future, and in particular he says this:
Towards the close of Treta and the beginning of Dwapara, I shall take birth as Rama, the son of Dasaratha in Iskshaku's royal line. At that time, the two Rishis viz., the two sons of Prajapati, called by the names of Ekata and Dwita, will in consequence of the injury done by them unto their brother Trita, have to take birth as apes, losing the beauty of the human form. Those apes that shall take birth in the race of Ekata and Dwita, shall become endued with great strength and mighty energy and will equal Sakra himself in prowess. All those apes, O regenerate one, will become my allies for accomplishing the business of the deities. I shall then slay the terrible lord of the Rakshasas, that wretch of Pulastya's race, viz., the fierce Ravana, that throne of all the worlds, together with all his children and followers.
There is information about Vibhishana and how he became immortal.
He asked for a boon that he should always be righteous and also obtain the light of divine knowledge from Brahma
Brahma then addressed Vibhishana, 'O my son, I am much pleased with thee! Ask any boon thou pleasest!' Thereupon, Vibhishana replied, 'Even in great danger, may I never swerve from the path of righteousness, and though ignorant, may I, O adorable Sire, be illumined with the light of divine knowledge!'
Brahma was so pleased by him that he even bestowed upon him immortality.
And Brahma replied, 'O scourge of thy enemies, as thy soul inclines not to unrighteousness although born in the Rakshasa race, I grant thee immortality!'
Sometimes, Krishna talked about Rama.
Similarly was the Rakshasa Ravana of Pulastya's race, with his relatives and followers, slain by Rama!
Krishna consoled Yudhisthira when he was in grief of the death of his kinsmen and told how Dasaratha's son Rama also fell prey to death.
We hear, O Srinjaya, that Rama also, the son of Dasaratha, fell a prey to death. He always cherished his subjects as if they were the sons of his own loins. In his dominions there were no widows and none that was helpless. Indeed, Rama in governing his kingdom always acted like his father Dasaratha. The clouds, yielding showers season ably, caused the crops to grow abundantly. During the period of his rule, food was always abundant in his kingdom. No death occurred by drowning or by fire. As long as Rama governed it, there was no fear in his kingdom of any disease. Every man lived for a thousand years, and every man was blessed with a thousand children. During the period of Rama's sway, all men were whole and all men attained the fruition of their wishes. The very women did not quarrel with one another, what need then be said of the men? During his rule his subjects were always devoted to virtue. Contented, crowned with fruition in respect of all the objects of their desire, fearless, free, and wedded to the vow of truth, were all the people when Rama governed the kingdom. The trees always bore flowers and fruit and were subject to no accidents. Every cow yielded milk filling a drona to the brim. Having dwelt, in the observance of severe penances, for four and ten years in the woods, Rama performed ten Horse-sacrifices of great splendour and to them the freest access was given to all. Possessed of youth, of a dark complexion, with red eyes, he looked like the leader of an elephantine herd. With aims stretching down to his knees and of handsome face, his shoulders were like those of a lion and the might of his arms great. Ascending upon the throne of Ayodhya, he ruled for ten thousand and ten hundred years. When, he O Srinjaya, who transcended thee in the four principal attributes and who was purer than thy son, fell a prey to death, do not grieve for thy son that is dead.
There is mention of even Valmiki, the author of Ramayana.
He was also present during the conversation between Yudhisthira and Bhishma which is recorded in Anushashana Parva of Mahabharata. For eg. In this chapter of Anushashana Parva, after Lord Krishna finishes reciting Mahadeva Sahasranaama to Yudhisthira and others, among with other sages, Valmiki also comments there as: Then the illustrious Valmiki, addressing Yudhishthira, said, --Once upon a time, in course of a dialectical disputation, certain ascetics that were possessors of the homa fire denounced me as one guilty of Brahmanicide. As soon as they had denounced me as such, the sin of Brahmanicide, O Bharata, possessed me. I then, for cleansing myself, sought the protection of the sinless Ishana who is irresistible in energy. I become cleansed of all my sins. That dispeller of all sorrows, viz., the destroyer of the triple city of the Asuras, said unto me,-Thy fame shall be great in the world.