So, with the polite form dying out among young people, is there any situation in which I (as a young person) would ever use the word "frøken"?
In some sentences Duolingo forces me to enter "frk." instead of "frøken", in some sentences it is the other way around. This is really puzzling me...
Hvad kan jeg hjælpe dem med, frøken?
The capitalisation of dem is the important part here. Since you also seem to learn German, it's just the same there: de/deres/dem (Da) and sie/ihr/ihnen (Ge) for they/their/them, and De/Deres/Dem (Da) and Sie/Ihr/Ihnen (Ge) for the formal you/your/you.
In English the polite form would be "what MAY I help you with", not "what can I help you with."
Ah, not exactly. These sentences have slightly different meanings, in both English and Danish. They are not different levels of politeness.
- What can I help you with? - Hvad kan jeg hjælpe Dem med? = Is there something within my skillset that I can do for you?
- What may I help you with? - Hvad må jeg hjælpe Dem med? = Is there something you allow me to do for you?
That's usually what English teachers tease you with if you say "Can I go to the bathroom?" or something. "Can" denotes the (physical or otherwise) ability to do something, "may" asks for a permission.