Create a custom terminal window is it possible to create (and put in the dock) a custom terminal window? 

What I mean with "custom":


it should have a different name (eg. "myTerminal");
it should automatically start running a certain command (I'll ask if this is possible in another question);


Any help is appreciated.
Yes, create a .terminal or .command file with the settings you like. The contents can then be a normal shell script that sets up the terminal the way you like it. Classic .sh files work as well, but the icon is not as nice. Place you file in a directory of your choice and add it to the dock.

Once you have that part working, you can probably paste a custom icon of your choice over the file using Finder and these posts.