Regarding the understanding of dialects, I wouldn't give much on the first impression of a German who only speak standard German. They usually don't have to un

Regarding the understanding of dialects, I wouldn't give much on the first impression of a German who only speak standard German. They usually don't have to understand dialects unless it's within the family context. So they have almost no practice in understanding German that is different from the standard and even slight accents can already throw them off. Compare that to Swiss or Austrian German where the dialects are used on every level of society and usually don't have a mark of being used by lower class citizens. It's not remotely strange to discuss quantum physics in Swiss German where Germans would most likely switch to standard German. In such a case, you can also look on how long it takes a native to understand the language without any formal education. In the case of Swiss German, I'd say it takes at most 6 months to understand everything (as usual YMMV as there are some peculiar dialects out there). But as you noticed with Danish<->German, even though there are some sight difference in grammar, there are some obvious transformations like "Haus" becomes "Huus" or "Zeit" becomes "Ziit" that can be generalized.