A: The Answer is NO
My Thoughts: WMT links show dofollow and nofollow, all the disavow tool does is tell google that some dofollows should be treated as nofollows. It does not tell Google to remove any links. The link count would only change if Google discovers those links and decides to deindex the page.
A: Within a couple of days to 1-2-3 weeks
My Thoughts: Recently it has been much faster, last year it took 28 days, now it seems to take between 3-8 days
A: Add them to disavow first then try to contact webmasters
My Thoughts: Matt Cutts was very specific last year saying completely the opposite, his video delayed my recover by almost 6 months! This clip below shows him giving wrong information to people. To this date I am unaware of Matt Cutts or any other Google spokesman publically denouncing the information given in this video. Very few people know of the "Office Hours Google+ Hangouts" so they would never have the chance to see the video clip of John Mueller telling you to do the opposite! A classic example of misinformation from Matt Cutts. There are too many to even mention over the years like this one, these comments have cost businesses untold millions.