Feel free to downvote this answer because it does not fulfill some of your requirements, but this is the way how I solved similar problem last year and I'm quite happy with it:
I decided to go online and take Exchange Online Plan 1. It is paid service $4/month and works like this:
(green=at Microsoft; blue=clients you are using)
So for most of the time I can have have my Outlook client (and even laptop) switched off and my mobile device (with Outlook Mobile, but iPhone etc. support Exchange, too) delivers me all my reminders and e-mails.
Outlook WebApp is a compromise between the two: almost full Outlook client is open inside the web page. (OWA should NOT to be confused with Outlook.com which has only basic functionalities.) It also plays notification sounds and has no taskbar button (because it is open in one of browser tabs).
The advantage is all 3 clients work independently, no connection between PC and mobile phone is needed. All the messaging takes up to 30 seconds. (Typically 10 seconds.)
Example: e-mail arrives to my mobile phone immediately after it was received by Exchange. Mobile phone is not checking server for e-mails (POP3, IMAP protocols), but Exchange server is actively notifying mobile phone when e-mail arrived (Push Mail – Exchange protocol).
If I update contact in mobile phone, in short time I find it updated in my desktop Outlook and vice versa.
If I'm without my laptop, I have full outlook at hand through Outlook WebApp.
For me this solution is definitely worth the money.
Mailbox size is 25 GB.
At my webhosting provider, I have redirected e-mails from my domain to Microsoft, but it works also without that: mailboxes have address firstname.lastname@example.org.
But after redirection of my own domain, email@example.com and firstname.lastname@example.org represent the same mailbox.
One more thing: technical support is included, so if connections stopped working etc. I got a call from Microsoft tech support representative (within 1-2 hours after reporting) working with me on the problem.
As a bonus you get access to some Exchange-only features like:
- e-mail Out-of-office assistant (sends Out-of-office replies)
- server-side Outlook rules – these rules work on server, so for example to sort some e-mails into folders or to delete them you do not need to start your Outlook. After starting Outlook you only find these actions are already done.
- e-mail retention policies – for example spam is assigned 90-days retention policy, all such messages are automatically deleted after 90 days
- remote deletion of your mobile device content – from administrative interface you can send "delete all data" command to your lost mobile device
- after you lose your Outlook storage or break entire laptop, you simply install Outlook again and all Outlook data are shortly back (including settings like rules etc.)