Going by this page,
Changing Button Functions on Mk320? - Logitech Forums, your hotkeys cannot be changed and they use your default programs for their actions. It does refer to your F-Keys but I assume it would be the same for the others as they supply no software to do this.
As I see it you have two options,
1. Use Tom's exe if we can get it to work with your key
2. Use a E-mail client, set as default (should work)
I recommend option 2.
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One problem I know with Tom's exe is if you click a e-mail address in a web browser it will take you to the login page or open another window/tab if your already logged in but you will then have to click create a message and manually copy the e-mail address that you clicked.
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I use Thunderbird and I know you can add Hotmail and GMail accounts to it. The limitations and security of this I don't know. I do have a Hotmail account setup in Thunderbird and my mother has a Gmail account setup and we both can recieve our messages and send e-mails with text, have not tried sending attachments or pictures inline.
The main benefit of using Thunderbird is when I click an address in my browser it opens a write message window in Thunderbird and I can choose from a drop down box the e-mail address I wish to send it from.
If you go with a E-mail client you will need to set it as the default e-mail program. If you have a different client already installed you can test your button before setting up any accounts in it. I also believe Windows Live client can use Hotmail and Gmail if you have that installed but I'm not sure.
If you want to go with Tom's exe the first thing I would try is open Default Progams from the Control Panel then,
Select "Set your default programs".
Select Outlook.
Click "Set this program as default".
The place you copy the Tom's exe to should require you to overwrite the original Outlook.exe (I would rename it to Outlook.exe.bak) so check that it's the right location. If your using
Office 2010 64bit it will be in
"C:\Program Files\Microsoft Office\Office14\OUTLOOK.exe"
32bit in "C:\Program Files (x86)\Microsoft Office\Office14\OUTLOOK.exe".
If the keyboard button still does not launch it try clicking a e-mail address in your web browser and let us know what happens.
Let us know how it goes or if you need more help with anything.