You can add your new alias before you move it.  Then when you move it the application start in the new location will pick up the new values.

To answer your question more directly though: Yes, you need to restart, and you can do that by updating the web.config file and resaving it. Just put a space in there somewhere that it doesn't matter.

The alias used will be whatever is used on the Url to access the site.

The alias is a way for the framework to distinguish between multiple portals.

You can have many aliases for a single portal and when you browse to a Url that hits the dotnetnuke app the framework will lookup and try to match a url to the alias, after it finds a match it will know which portal you are requesting and load the home page for that portal.

No you don't have to worry about the old alias being in there, but if it is not valid you should remove it.  Since there is another feature that happens when the Url can not be matched to any alias.  If this happens the framework will redirect to the first alias.

More info can be found here: DotNetNuke Portal Alias explained