Shirin you made as sugar for Perwiz
Ferhad shed blood as tears
The poet Nizami Ganjavi, who lived in what is now Azerbaijan in the 12th Century, wrote a poem 'Khosrow o Shirin' which is based on a pre-Islamic story from Persian historical epics based on real events.
In this story a Persian king, Khosrow II Parviz (I assume this last name is the same name as 'Perwiz) courts the Armenian Princess Shirin.
The story seems to have farcical elements in that the couple keep missing, or failing to recognise each other as they try to travel to meet up. They finally meet and she agrees to marry him if he reclaims his country from an enemy, his efforts to do that result in him having to marry someone else.
Meanwhile a sculptor, Farhad falls for Shirin, and so Khosrow sets Farhad impossible tasks in distant places to get him away from her. Then Khosrow tells Farhad that Shirin is dead, so Farhad throws himself off a mountain to his death.
Then Khosrow's wife dies , possibly poisoned by Shirin. Before rushing off to marry Shirin, Khosrow takes time out to try and get his leg over with another woman.
When he finally tips up at Shirin's castle he's drunk as a skunk and she tells him to go home.
Eventually they marry, but Khosrow's son from that awkward 'other marriage' shows up and falls in love with his step-mother. So he kills his Dad and sends Shirin a note that he's king now and she has to marry him. She goes 'Yeah, right' and kills herself.
(summarised from the Wikipedia entry)
Layla you made an affliction for Qays
A poem about Layla and Qays was also written by Nizami Ganjavi, although the story is apparently of Arabic origin, later absorbed into Persian culture.
From Daily Sabah the story is summarised:
Layla and Qays (Kays) fall in love with each other at first sight in their childhood. Qays is so sure of this love and begins to write love poems for his lover. His fondness for Layla causes people who hear about their love to refer to him as “Majnun,” which means "crazy" in Arabic. As knowledge of their love spreads around the community, it becomes impossible to hide it from their parents. Upon this, Qays finds the courage to ask Layla’s father for her hand in marriage. However, the father refuses by informing the young man that his daughter cannot marry a guy named "crazy."
With imponderable grief, Majnun disappears in the deserts by living an isolated life among wild animals. In his time in the wilderness, Majnun replaces his material love for Leyla with divine love and becomes a hermit. Layla is also miserable and is forced to marry another guy. After her husband’s death some years later, Layla hopes to be with her only love again and goes to find Majnun. When she finds him, Majnun does not recognize her material being. It is then that Layla understands they will never meet again in this world. Finally, she passes away with a broken heart. Hearing her knell, Majnun comes to her grave and cries over it. Mourning to God to take him as well, Majnun also dies soon after.
One of the loyal friends of Majnun sees the couple in his dream later on. In the dream, they are in the garden of heaven together, symbolizing that the duo reunited outside the confines of time and the material world.
Ramin you entangled with Ways
Vis (Ways) and Rāmin is another classical Persian romance. The poet who composed it as an epic in the 11th Century claimed it was of Sasanian origin (The Sassanid Empire being the last Persian imperial dynasty before the Muslim conquest) though is is now thought to be Parthian in origin.
The pair grow up together in the care of a wet nurse. She is the daughter of a ruling family in Western Iran, he is the brother of a king in Northern Iran.
At adolescence they return to their families.
Vis is married off to her brother (or possibly her Uncle, the wikipedia synopsis is ambiguous) despite having been promised to Rāmin's brother (who had wanted to marry her mother and got offered Vis as a consolation prize).
Much war and killing ensues, during which Vis and Rāmin get around to falling in love. Eventually he ends up with his brother's throne, they marry, she dies when he's been king for 81 years and he mourns at her tomb for 2 years then dies.
Why did you show Yousif to Zulaikha?
Yousif and Zulaikha as, as Peter Shor notes in comments, appears in the Old Testament where she is the wife of the Pharaoh's Guard, Potipher.
She tries to seduce Joseph and accuses him of attacking her when she is rebuffed. In the Old Testament her version is believed and Joseph imprisoned. In the Qur'an the evidence reveals that it was her scheme, but as far as I can make out he still got imprisoned for being too damn gorgeous and not doing what he was told.
Some Jewish interpretations say that Zuleikha was merely confused by an astrologer being one generation adrift in predicting who in the Potiphar family was going to have Joseph's babies. It was Zuleikha's daughter who did that in the end.
How did you lead Wamik to Ezra
I can find references to there being a romance of Wamiq and Azra, but can find next to no information about it beyond that it may have been written in imitation of the poet Nizami Ganjavi's telling of the Khosrow and Shirin tale.
The Sheikh and Nilufer
It is worth drawing attention to the fact that the footnote in the translated source does not refer to all of these people as famous lovers:
Shirin and Perwiz, Ferhad and Layla, Qays, Ramin and Ways, Yousif and Zulaikha, Wamik and Ezra : famous lovers.
Leaving the Sheikh and Nilufer in a slighly more ambiguous position. A further footnote after the word Sheikh states:
Sanaani : the Pious who fell in love with the daughter of the King of Armenia.
I've been unable to identify any sheikhs of note who might have fallen in love with daughters of kings or Armenia. Armenia has had a number of kings and with several wives apiece, collectively acquired quite a lot of daughters. Added to that, 'Sanaani' appears to be the name of a variety of Arabic, as well as, or rather than, a name so that aspect of the footnote didn't help with narrowing down the field, not frankly did references to the pious and the impious.
The most likely candidate for 'Nilufer the tender flower', in terms of being pretty much the only person of the appropriate period who comes up in searches, is Nilüfer Hatun, first wife of Orhan, the second sultan of the Ottoman Empire.
The wikipedia article about her is pretty confusing, but it seems that while it is known who she was married to and who she was mother to, there is less certainty about her origins and how the marriage came about.
She may have been
daughter of the Byzantine ruler (Tekfur) of Bilecik, called Holofira. As some stories go, Orhan's father Osman raided Bilecik at the time of Holofira's wedding arriving there with rich presents and disguised and hidden soldiers. Holofira was among the loot and given to Orhan.
Modern researchers doubt this story:
Other historians make her a daughter of the Prince of Yarhisar or a Byzantine Princess Helen (Nilüfer), who was of ethnic Greek descent.
The wedding abduction story has her converting to islam and adopting the name Nilufer in place of 'Holofira'. One might speculate that despite being married via abduction she took to her new husband and became 'a lover'.
Note though that she is not a daughter of an Armenian King, so far as wikipedia relates at least, so I'm confident that the sheikh and her, as they each get a couplet to themselves, are not to be read as related.