Both of these are possible. If your pre-installed mail app won't do it, you can install one from the Play Store that will. I use Kaiten Mail as my email client, which is a paid version of the free K9 Mail app, so am using this as an example, but there will probably be others that will let you do this too.
If I open Kaiten and look at its settings, it has Account Settings with Notifications as one of the options. So I can set different notifications for different accounts, and hence could have one with an audible notification and one with a silent notification or none.
I've only got one email account set up on my phone (I don't want my work accounts on there), so can't test this, but I just tried adding a shortcut to my desktop and one of the options was "Kaiten Mail Accounts", and selecting that let me choose to set an icon which took me directly to a specific account's inbox or the unified (all accounts together) inbox. So I figure that if I set up a second account I could also have separate icons for each.
If you use a third-party launcher as well you could also edit the icons after creating them so that the two look different as well as taking you to different inboxes. Some launchers (e.g. Nova or Apex) can also add badges which tell you how many unread mails there are (with an add-on in the case of those 2 launchers).