Free email for my domain name e.g. [email protected]?

You will need to register that domain name and then have an email provider. It is unlikely that you will be able to find a company that allows you to register a top level domain name for free because they would have to pay the domain name registration fee. However you can get a free subdomain name with free e-mail hosting at http://www.subdomain.com/. It works by setting up a subdomain name like http://mydomain.com.nu (where the nu extension is added to your domain name).
If you must have a top level domain name you will have to pay a fee. However an easy to use and quick domain hosting service can be found at NetworkSolutions.com.
A third option would be to host the website yourself on one of your own computers. DynDns (http://www.dyndns.com/) can dynamically forward requests for a domain name to your computer. You can either register a domain name from them (for a fee) or use a subdomain name (like mydomain.selfip.com) as your domain name for free. Then you could use exchange your another email hosting service to host your email.

purchase a web site call (you may get one for as low as $2/12 months) and web hosting (for as low as $4/month) and you have your very own digital mail handle, area blanketed. (purely have the mail forwarded on your common account - no person will comprehend.)

Do you already have "mydomain.com"? If yes, then talk to your domain registrar/web host and see if they offer any email services. If they don't then you might need to find a web host who does.

If you don't already own "mydomain.com" you will need to register it first. Many domain registrars/hosts offer email services as part of their package, so you will be able to create [email protected]. If you want inexpensive domain registration/email try Doteasy.com (they are having a Pre-Boxing Day Sale) or GoDaddy.com.