Why isn't the 'a' included: "Hun har en kjole og sko pa"
So sko is also the same single as plural, just like "dyr" and "mus"?
Yes. en sko, flere sko, alle skoene
Does "Hun har på sig kjole og sko" sound wrong? I would say it means the same
Yes it does sound wrong. It would be like saying "She has on her a dress and shoes." The sentence doesn't differentiate between her having the clothes on or in her arms.
Just wondering if somebody can clarify what the purpose of 'på' is!
Compare it to the English sentence: "She has a dress and shoes on". "På" works in the same way "on" does in the English sentence.
Why is there not "en/et" in front of "kjole"?
See Xneb's comment above: 'In Danish, if you are saying you have "a(n) X" on, you would just say "Jeg har X på"'