Hi Vladmir
Perhaps I can help a little...
1. It depends on the type of duplicate content. Three scenarios;
-- two pages that are exactly, or almost exactly, alike. this means titles, on-page content etc. with those, you really should 301 redirect one to the other OR noindex one of them. then submit a URL removal request in WMT
-- pages that only have duplicate title tags and/or meta descriptions. assuming the on-page content is different and its just the title tags that are duplicate, change the title tags so they are unique. or, if the pages don't belong in the index (like paginated pages) noindex them.
-- lastly, you can have the same page_but it appears duplicate because of parameters like /search.php?search=apples - the page is 'search.php' but the parameters make it look different each time. This is the most appropriate time to just add a canonical._
2. In WMT, first submit a URL to 'crawl as Googlebot' - then after that crawl you will have an option to submit the URL to Google. This should get them to see the change sooner. Even if you didn't do this, any normal page should get re-cached in less than a month
3. Assuming we're just talking about on-page changes, this is also when they re-cache the page. That's when you should see the biggest change.
Hope that helps out!
-Dan