Regarding the names of the months in the Danish Outlook 2016 (I thinks all versions actually) you're right that the name of a month shouldn't get capitalized. However when it starts a sentence or is used as a heading, it actually should be written like "April 2019" and not "april 2019" like in the screen dump below.
Maybe it's not doable since you're are using same strings for "1-7. april 2019", but now you have been advised ;)
The same is the case regarding local weather where "lørdag" should be "Lørdag" as it is being used as a heading.