What's the best way to configure my keyboard so I'm not coming up to the mouse every time I need an umlaut?
Just type "e" after the letter with the umlaut! That's how umlauts are spelled out when you don't have access to the correct keyboard (my wife writes like this when emailing her parents from her USA laptop, etc). Ü is ue, Ö is oe, Ä is ae. Duolingo accepts this without any problems.
On a mac, Option+U and then the letter you want will create an Umlauted letter. For Windows, you have to hold down ALT while typing a code for the letter you want, but it's easier to just use the mouse. The code are: Ä 0196, Ë 0203, Ï 0207, Ö 0214, Ü 0220, Ÿ 0159.
In X, you can use xmodmap to add AltGr and dead-key combinations. In Windows, you can do the Alt+code method, or you can download a driver to enable AltGr (I have a Dvorak+German driver for those rare occasions that I boot into Windows). For other operating/windowing systems, check the documentation.
How about the Eszett in Windows keyboards? Any comfortable shortcut or way to customize its position (Ctrl+S, Alt+S or similar)? It is annoying seeing the red notice "Be careful with..." everytime I type a "ss".