For example: When American Dad is aired in Germany in German are the originally English-language songs such as "Is She Not Hot Enough" and "We've Been Bad" (Krampus and Steve's duet) sung in German?
In general, no. The songs remain in English, using the original voice track rather than rerecordings. They only add subtitles, often with a very rough 1:1 translation.
Also this is only done for original songs that are longer than just a few words. For example, Roger or Steve walking through a scene humming or singing a few words won't receive subtitles, if it's from some popular song and not original.
Overall, this isn't some general, unwritten rule or anything like that. Whether songs are translated and/or rerecorded often varies from show to show and sometimes even episode to episode (e.g. The Simpsons have many translated songs but also many that are only dubbed (in a really bad way).
Also keep in mind that many of these shows unfortunately don't have a prime time spot in Germany. Actually only The Simpsons airs new episodes at prime time, other series (like American Dad, Family Guy, etc.) are stuck on odd times or smaller, less popular stations, so they often have a smaller budget and getting songs translated right is often significantly more work (and therefore more expensive; I'm working as a translator).