As far as I know, classic Chinese is not used to "write" these languages as you think. Classic Chinese was just used as an "international" language among surrounding countries, like English nowadays. Take the Japanese Language for example.
In ancient times, the Japanese had their own language, but they didn't have a writing system. Of course, China was powerful back then and exerted influence over the surrounding countries. Japan also imported Chinese characters and culture, especially during the beginning of the Tang Dynasty (唐朝) in China.
After learning Chinese characters, some Japanese scholars started to use characters to note the pronuciations for Japanese words, something like the PinYin system, as now we use Latin letters express Chinese. However, every character had its own pronunciation and meaning, so the Japanese created their own "letters", now called "hiragana" and "katakana", based on Chinese characters. For example, the "hiragana" "あ" comes from the character "安", "か" from "加". They look similar, right?
This also happened to Vietnamese. The Vietnamese created their Vietnamese characters(喃字) based on Chinese characters.
Nowadays, the Koreans don't use Chinese characters(but some characters are still used, especially in solemn situation and law), the Japanese still use them, and the Vietnamese quit thoroughly because the French instituted the use of a Roman alphabet during the colonization period.
So I think the answers to your questions would be:
- None. They developed their own writing system based on Chinese characters and borrowed a lot of words from Chinese
- Since characters are used as writing system and letters to note the pronunciation. Grammar is not the point
- Meanings for some characters changes through time. Furthermore, grammar also changes, so if you want to learn classic Chinese, you'd better learn mordern Chinese first. After you learn mordern Chinese, it won't be so hard for you to learn classic Chinese
I suggest everyone who wants to learn Chinese: you might as well learn some classic Chinese. Although we don't speak in Classic Chinese, there are many words(especially idioms) from classic Chinese, such as "多行不义必自毙"，“退避三舍”,“秋波”。Also, you can read some classic poems to get a feeling of classic Chinese.
Edit: More explanations on point 2.
To clarify how the claiss Chinese were imported and affected Japanese. Imagine the steps below.
STEP 1: Japanese had their language, so they could say: "この山は 高い" (This mountain is high), but they could not write this sentence.
STEP 2: They got to learn classic Chinese, and they(especially the nobels) decided to use classic Chinese for communication, because Chinese culture was so attractive to them. Now the classic Chinese worked as a popular foreign language, like English nowadays.
STEP 3: Some Japanese scholars thought:"well, Chinese seems good, why not create our writing system based on it?", and eventually they invented the "hiragara" and "katakara", by abstracting the Chinese Characters. For example, now they have "こ" from “己” and "の" from "乃“ and so on.
STEP 4: Now they could use these "letters" to write down the sentence. It could be like this(without any characters): この やま は、たかい, but it's not so conevnient because they had to divide every word in writing.
STEP 5: "Hey, we have learnt Chinese. In Chinese, “やま” is "山", and character "高" means high, in our language, it's たかい. Why not use characters to simplify the writing system?", Japanese might think this way. Of course, in Japanese, an adjective has serveral kinds of conjugations, but that doesn't matter, because only the suffix(い) should change, the stem(たか) won't change, so they just used the character “高” for the stem and everything works well. I.e. the negative form of "高い" is "高くない".
So I think you understand why importing characters won't affect the grammar.