Don't add any code directly to layout.master, create a UserControl ie a .ascx and embed that in layout.master then your logic can be inside your usercontrol.

you can do whatever you want in your own code.

myself I don't access the db directly from the UI layer. I have the UI talk to a business layer class which talks to a data access class and the data access class would get its connection string from a config setting. so yes you would need a separate config setting for your different connection string. you could put all your code in the usercontrol but that is not how I do things myself.