What you are looking for, is a Blackletter or Gothic font, such as these:
They span from the medieval to the renaissance, but are probably generally seen as being medieval rather than renaissance. There are hundreds of fonts like these, you will find a lot here and here.
The Renaissance was roughly from the 14th to the 17th century, but Renaissance meaning "rebirth" it is often referred to the latter part, as the influence of "classical" arts became more pronounced.
Seems to me you are more interested in the late medieval.
Which one you should use? Much dependant on context. I would personally suggest you stay clear of Old english, as it is so overused it is painful.
Your second example seems to me a perfectly good option. The one in the logo makes my eyes hurt from all the clichés piled up; a very last-century-design. But that is my personal opinion. But it is all about how you treat the font. Just make sure it is reasonably readable.
I think your confusion with adding "scandinavian" might come from a misunderstanding that these kind of fonts are often used by scandinavian heavy metal bands, but there is no cultural link there at all.
Edit: regarding Denmark: you could then focus on german blackletter as that would be the closest cultural connection you will find.