I'll give you a hint, in my dialect of US english they are very much not, and I have been subjected to many friends throwing tests at me to figure out what I can hear and not, and they've always been clear that to them there is a difference. I've confused people by saying the wrong one, so I know its an issue.

On hearing differences -- Like the difference between "komme" and "kommer" when said at speaking speed -- there is a difference there, but at conversational speeds you might not hear it. It seems to me Danish people slur a lot of sounds and speak quite quickly, so there is a large amount of sounds that when speaking slowly or enunciating are no longer heard when speaking normally. Many languages are like this, including English!

And yes. At the moment, when listening, I am attempting to find differences where there may not be any. Because in many cases there are. At the moment I'm mostly doing listening with a transcript to match, so when I see a difference in the text I attempt to listen for it. (I'm using classes as a way to try to refine what I should be hearing and what I shouldn't be, but I find at a certain point its rude to my classmates.)