1. Would the version ii 70-200 work with version ii converters
Yes - in fact the iii extender became available well after the lens was released.
2. Would the version ii converters work better with the 70-200 mark I
Not 100% sure on that, but I believe they did make some optical improvements so I'd expect so.
3. Is it important to purchase mark iii converters for the 70-200 ii
The mark iii has weather sealing and is a little better optically. It has some AF speed improvements with newer super-teles but I don't think those changes make a difference on the 70-200. Guess it's up to you if the extra $$$ are worth a small improvement, but either will work fine.
As for your main question I think the 70-200 II is considered to have better IQ than the 70-200 I in just about all regards, so the same will apply when using an extender even more so. The II version also has faster autofocus which will help when using an extender, which slows down AF.