I thought this might be the case. Thanks. What I understand, from your explanation and example, is that the two vowels are not pronounced in two, separate syllables, but rather as a diphthong would be in English. That's sort of what's happening in "tights" where one vowel really has two vowel sounds that glide from one to the other. I don't know if Russian is supposed to officially have diphthongs, but the "ае" in китаец gets pretty close to the same thing and when said fast they apparently glide together and become indistinguishable from the shwa sound.
Exactly. I'll give you a couple of other examples. The word загородный in which the stress falls on the first syllable and which means "suburban" or "rural" is pronounced close to ZUGgurdny. The word недопонимание (lack of understanding) in which "a" is stressed is pronounced close to "need a pnim MAHN-ye" in a fast speech. The general trend is to emphasize the stressed vowel and the one in the preceding syllable and make all other vowels as vague and short as possible. By the way, in Russian words shwa never occurs in positions preceding the stressed syllable. For that reason Russians have difficulty pronouncing the indefinite article "a" correctly. For instance, lots of Russians say "eh book" or "uh book" instead of "a book".
It is probably because of the spelling mistake (cheacher vs teacher) or (more likely) because the teacher’s sex is irrelevant. What matters is that the teacher comes from China. «Наша учительница - китаянка» is also translated as “Our teacher is Chinese”, but one difference between Russian and English is that in Russian you can hardly mention someone’s occupation without explicitly stating the person’s sex. This feature presents a challenge to English-Russian interpreters who have to ask the speaker to mention the gender of every person they are talking about.