I'll try to keep this short but you're asking for some elaborate answers that are difficult to articulate.
Are they reassigning every note of the scale to those assignments in the other scale?
This the easier one to address so we'll start here. The answer is yes.
As you know, major and minor scales are both seven notes each. The difference between them is that that third, sixth and seventh step on the major scale are one note lower on the equivalent minor scale.
Major: 0 2 4 5 7 9 11 (12)
Minor: 0 2 3 5 7 8 10 (12)
If a song is in D Major, it would be...
Major: D E F# G A B C# (D)
To make it D Minor, those three notes get transposed by a half-step.
Minor: D E F G A A# C (D)
Why does it sound good to use a minor key for a song that is conventionally done in a major key?
Listening preferences are almost entirely environmental. You build your taste in music on things that you've heard and have associated certain experiences with, whether or not you know it. At some point, you heard this piece of music and your brain lit up in some way, and you developed a connection between that particular sound and a particularly important feeling. Over time, you refine that attachment to become more and more specific.
To really address this point, we have to look at how memory tends to function optimally when there's an emotional impact. You can likely remember what you were doing right before you learned that your grandmother passed away but you probably don't know what you had for dinner last Tuesday. The brain usually recognizes when things are mundane and not worth keeping a record of.
Songs built on minor keys tend to be more moody, whereas major keys sound more uplifting and "fun" (for lack of a better word). However, major keys can also be used for a reminiscent emotion when they're applied just right.
For example, compare how heavy In The Air Tonight feels against the way that In The Arms Of Angel invokes a more "defeated" feeling. Music is great at illustrating the abstract.
Case in point, we're constantly chasing extreme moments. Work is boring. TV is boring. You want to feel something (for reasons that would require another wall-of-text) and music has a way of triggering the right extreme emotions. The appeal of hearing something in a minor key is that it feels more "serious", and you're wired to associate that sound with a sense of importance.
You know how the Star Spangled Banner sounds, but when it's played in a minor key, your brain catches a rare moment of that extreme emotion. It says, "Oh, shit. I understand this but it's different and you should savor this experience of hearing it for the first time."
The joke is funny. What is the practical purpose of doing that at all? That's not the song.
Humor is an entirely different discussion for another day.