From what I've read, the last Sassanid king,Yazdegerd III, went to China Tang with his court after his realm was conquered by the rising Muslims. He didn't make it, but Peroz III, Yazdegerd's son, did. The Chinese helped him to set up a governorate/march around Afghanistan. Something happened and his entire court then settled in China for good, with Peroz's son, Narsieh seemingly assimilated and even adopted the family name Li.
Some questions on these events:
Why did the Persians sought to flee to China? Were they confident the Chinese would help them? What is the rationale for these actions or thoughts?
Why were the Chinese so welcoming to the point of helping Peroz set up a far flung governorate? Was it an altruistic decision? Some historical ties? Favour for a deposed prince? Expansion of sphere of influence?
Why did Peroz finally settled in China?
How was it for him and his people living in China? Were they treated as refugees/exiles?
What was the Chinese's reaction to Narsieh adopting the imperial family's family name? What happened to the House of Sasan after these events?
This is a very obscure topic I have only ever found two-three secondary sources discussing (there are probably more in Chinese). Unfortunately, I don't think most of your questions are answerable; there's not much material on it, scarcely a page in Touraj Daryaee's 2013 textbook on Sasanian Iran.
So to go over them in turn:
We don't know, except that the Chinese recognized the royal status of the Persian Royal Family. My speculation would be that the Chinese acknowledged the divine mandate of the Persian kings - they did refer to Zoroastrianism as "Heaven Worship", after all.
See 1; they had diplomatic ties. Sphere of influence must have been a consideration as well.
"Probably because of further Arab Muslim victories", per Daryaee.
Their royal status was acknowledged and they pursued careers as military commanders. A statue still stands outside the mausoleum of Gaozong with the inscription, "Peroz, King of Perisa, Grand General of the Right Courageous Guard, Commander-in-Chief of Persia".
No idea what the reaction to the surname adoption was; I assume it must tie into Chinese custom. An expert of China will probably be able to help. Peroz's brother Wahram, and his son Khusro, both attempted to invade Persia, but were ultimately unsuccessful, and that's the last we hear of the family as such - though of course, many would go on to claim descent from them.
As you can guess, there's not a ton of material to go on, so this is the best I can do.