I can recommend Thunderbird.
- cross-platform (GNU/Linux, Mac OS X, Microsoft Windows),
- Free/Libre Open Source,
- gratis available
email client from Mozilla.
Managing multiple accounts is easy and works great. Each account has its own settings, and each account can have several identities/aliases (each with its own email address, name, and signature), with one being the default/main address for this account.
All accounts are listed in a sidebar, in a folder tree (with all their email folders as children). You can switch to different views, e.g., showing only email folders with unread emails.
I’m using Thunderbird for many years with +/- 15 accounts and 40 email addresses. Works like a charm.
When composing an email
From field becomes a select list, where you can choose which email address to use. It displays the chosen name and the email address for each entry. Aliases are grouped under their accounts.
If you compose a new email while being in a specific account (e.g., browsing its folders), the account’s default email address is automatically selected. The same is the case when answering an email (it selects the email address the original email was sent to).
When a new email arrives
You can disable overlay notifications under Options → General. After this, the only times you should see pop-ups are cases of warnings/errors, e.g., if it can’t connect to an account.
By default, you can enable playing a sound only globally, i.e., for all accounts (there’s an old feature request to change this). There are some add-ons that allow to customize this:
I can’t recommend one, as I’ve never used audio notifications. Judging from their descriptions, ToneQuilla seems to be suitable for your case: create a filter that fires when your selected email addresses are used in the
Windows 7 integration: minimize to task bar
The add-on Minimize On Start and Close (which I have never used) seems to make it possible for Windows 7 (emphasis mine):
Instead of minimize the window to the tray it minimize it to the taskbar. It can minimize on startup, when the window is closed and/or when the esc key is pressed.
Related wiki entry, maybe of interest: Minimize to system tray.
For questions about using/configuring it, Super User seems to be a good place, with more than 1000 Thunderbird questions.