I'd recommend that you consider instead only supporting IE 10 and/or 11, and prompting users of lower versions to upgrade to any supported browser (e.g. Firefox/Chrome for users who can't upgrade Windows, and IE 10/11 for users on Windows 7/8).
Edit: You also want to think about correlation vs. causation -- is it the choice of browser that causes a user to be more engaged, or is is that more engaged users are more likely to choose Chrome for an external reason? (e.g. because they're Google fanboys, or because they tend to have a hundred tabs open, or because they need account sync because they use three different computers, etc.)
Safari is the primary client of choice for iPhone, iPad and most Mac users. Discontinuing support for them basically cuts off the majority of the Apple ecosystem.
As far as IE goes, There are a few things to consider. Are you spending more time in IE development then you feel you should? When you look at your IE breakdown, could you shave hours of development time by dropping support for IE 6, 7?
There are some limits you should be aware of, if a lot of your OS traffic is still on Windows XP, those users can't upgrade to IE 9. Chances are, they are less likely to switch if they are -still- on Windows XP.
Really, if you are developing standards based HTML, supporting Safari and IE shouldn't drastically increase your development time (provided your not supporting IE 6/7). So you stand a chance to lose 12% of traffic... for what really?
As far as, your second question. You could track outbound links to the download page for Chrome, and you could store the User Agent for registered users and do a check, however it's not very clear why you want to do this in the first place.
I think, if you go forward with this, you will find that most of that 12% won't switch and will either keep using the site as is, or just stop visiting.
Lastly, Chrome 28 switches to the Blink rendering engine. This means you wouldn't be checking your site against: Gecko, Trident or Webkit? Do you really feel that is the best solution?