Yes you can fire the KeyDown or KeyUp event on a TextBox. To check whether it was the enter key, you can do the following:
//e is the KeyEventArgs from the event.
e.KeyCode == Keys.Enter
Then, if he has pressed the enter key, you can do:
System.Windows.Forms.Control.SelectNextControl();
To set the order of your controls, in Visual Studio look for this little icon:
http://i.stack.imgur.com/nZWLO.png
Click it, and you'll go into tab ordering mode, as I like to call it. Just click the controls in the order you wish them to be and after you're done, click the little icon again. Presto!
Now whether that is a good idea of not, completely depends on how used to it your end users are. If they have always used it like this, and you give them something that doesn't fit into their mind model, they are going to say your software is broken.
Always always always try to emulate what processes the user already has in place in their head.
Read this if you have the time, it's a really light and very good read:
http://www.joelonsoftware.com/uibook/fog0000000249.html