The answer is no; we cannot take the phrase
the well-educated chemistry teacher with Chinese eyes
and stick the possessive morpheme on it.
It's completely out of the question in writing (where that morpheme becomes apostrophe-s that ends up on the wrong noun). What you are doing there is ineffective and confusing even in informal writing, such as an e-mail between friends.
In spoken English, it's still extremely dodgy. Kindergarten age children do this sort of thing: "that's yucky, don't touch it; it's that dog with the big ears's chewing ball".
You can always put the possessive morpheme on noun phrases whose root noun appears in the rightmost position, where it then appears on the correct noun.
To get the noun in the rightmost position requires you to "refactor" the additional prepositional phrases into adjective phrases. For instance "with Chinese eyes" might become "Chinese-eyed".
the well-educated, Chinese-eyed chemistry teacher's computer
English-speaking kids learn this through correction. "Johnny, we don't say dog with the big 'earzes' ball! Try this: the big-eared dog's ball, hmm?"
Or else you just have to use "of". Using "of" for simple material belonging, however, is awkward. We rarely ever say "the computer of John" rather than "John's computer". It's not natural. Consider the alternative "belonging to".
The computer belonging to the well-educated chemistry teacher with the Chinese eyes.
However, please don't interpret this the wrong way: "X of Y" is used for simple belonging. Whether it is appropriate depends on the context. I won't get into it in this answer, but consider for instance that "I ran into Mrs. Hendry, wife of John Hendry, at the library" is fine, but "I heard the wife of John Hendry left him" is unnatural would be expressed as "I heard John Hendry's wife left him".