I’m curious about somebody with lifelong hearing deficiency, not somebody who loses their hearing due to age. What would communication have been like? Is there any historical record of deaf rulers? Was it treated differently in Europe vs Asia? How long would it take them to notice a baby or child is deaf?
There is a fairly large gap in historical knowledge of deaf people around this time, so answers to your questions are hard to come by. It is likely that deaf people were thought to be mentally disabled as well, and St. Augustine taught that deaf children were born as punishment for their parents' sins. For those deaf people who survived to adulthood, they were likely either kept at home by their families, or homeless - there were probably very few who were welcomed as productive members of society.
The only historic ruler of any kind with hearing loss that I know of would be Emperor Hadrian, who is rumored to have been hard of hearing. The only deaf person at all from the Middle Ages whose name we know (sort of) is the "Dumb Boy of Hexham" - reportedly St. John of Beverly taught him to speak.
Again, there is a major gap in information about European deaf people at the time; I know of no resources for Asia in this period. Even into the 20th century, children would not be identified as deaf until they were three years old or even later, so it is likely that identification in the Middle Ages was around the same time.
Sorry to give so much speculation and no sources - I feel a bit odd giving no sources, but there just aren't really any answers to your questions based on current knowledge.
I don't actually know the answer to this myself, but I do know there are some medieval historians working on disabilities in the Middle Ages, from literary, medical, and legal points of view. Notably there is Jonathan Hsy at George Washington University - according to his faculty page he is working on a book called Disability and Life Writing: Authorship as Advocacy, Then and Now, but it might be awhile before it's published.
Dr. Hsy has also written “Analytical Survey: Encountering Disability in the Middle Ages” (with Richard H. Godden). New Medieval Literatures, Vol. 15 (for 2013), eds. Laura Ashe, David Lawton, and Wendy Scase (Brepols, 2015), pp. 313-339.
The other best place to look would be Irina Metzler, Disability in Medieval Europe: Thinking about physical impairment during the high Middle Ages, c. 1100-1400 (London: Routledge, 2013). Another article worth mentioning (and which is cited in Metzler's book) is Sharon Farmer, "A Deaf-Mute's Story", in Miri Rubin (ed.), Medieval Christianity in Practice (Princeton University Press, 2009), 203–9.
Disability Studies is a fairly new kind of historical research, especially for the Middle Ages when the sources are pretty scarce, so there isn't really much to go on yet. But hopefully this is helpful.