I am studying the writing style of Leo Tolstoy by reading an English translation of Anna Karenina.
As a general rule, do translations affect the original style? In what ways?
When books get translated into another language, the writing style changes as well. Some literary devices in the source language just don't work in the target language for whatever reason. One person once asked here on the Stack Exchange about a Russian-to-English translation and said how Russian texts could have present tense and past tense, but this literary device was not translatable into English. So, the translator had to figure out another way to convey the same feeling. A good translator would adjust the story in order to appeal to a new audience. When you are translating between languages, you are not only translating the language, you are also translating the entire culture, making the people inside the culture seem human-like and real so that people in the new culture can identify with the characters. This video explains that translating Harry Potter into other languages was a nightmare, because the translator had to deal with numerous cultural references in Great Britain, British slang and colloquialisms and regional dialects, and English-based puns.
As an English-Mandarin bilingual, I can speak anecdotally that stories written in Chinese feel different from stories written in English. A sentence-level translation probably wouldn't work, because Chinese idioms and puns don't translate so well into English. Puns are notoriously difficult to translate, and Chinese idiomatic expressions may be used to describe a situation as an adjective/adverb. You can find an equivalent English idiom to match the Chinese idiom, but the problem is, your English reader would probably find that English idiom a cliché. Chinese idioms may also feel more structured or patterned. For example:
These idioms are not directly translatable into English. I mean, you can translate the meanings across fine, but the result would be ugly and unpoetic, doing a disservice to the original prose. Also, word choice also matters in Chinese. In English, the author has to consider word choice too, but it's different.
In Chinese, you can say 绿色的 or 绿油油的 to describe something as "green". What is the difference? 绿色的 sounds like a statement of fact. Boring, dry. It is what it is. 绿油油的 sounds more poetic or cute. Great for describing the beauty of green leaves. In English, you can use "green" and "verdant" to describe the same color, but note that the color is not so obvious in the word verdant.
I highly recommend Douglas Hofstadter's Le Ton Beau De Marot: In Praise Of The Music Of Language which deals in depth with exactly this question.
In short: yes! There are endless decisions to be made in translating between one language and another, making the job of a translator extremely complex. They are artists and writers themselves, not mere transcribers.
If you want to capture Tolstoy's "style" without actually reading Russian, your best bet is probably to find a number of different translations of his book, to compare/contrast.