I can only say "no" to that one. There is no difference between logical and grammatical emphasis here. "Do" is a handle which manipulates the verb phrase outside of simple affirmative sentences, and the verb phrase is "speak [language]".

Simple affirmative: she speaks [language].
Question: does she speak [language]?
Negative: She doesn't speak [language].
Emphatic affirmative: she does speak [language].

The difference you imply with your example "She doesn't speak English but she does write English.", which seems to state that somehow the emphasis is restricted to the verb itself and not the verb phrase, so that you'd expect the verb to change, does not exist in English. In fact, The stub sentence "she doesn't speak English, but she does" up to this point leaves it open whether the negated element is only the verb, as in your example, or is the whole verb phrase, as in cepo's example. Thats why you have to repeat both the verb and the language; you can neither say "but she does write." without at least adding "it", nor "but she does English." without repeating the verb (OK, that would probably be possible in colloquial speech, but not in writing).

And your alternative sentence "It is English that she doesn't speak but it is Russian that she speaks." even with the third person -s is extremely awkward and not a sentence you'd find commonly formed by any native speaker.