My understanding is that var is where, but vart is where to (or whither, if you're fancy). So "Var simmar anken" might have answers like "In the pond", but "Vart simmar anken" would want a destination, like "back to her nest".
If "var" is like "wo" in germam for where (position) and "vart" is like "wohin" (where to), what is the equivalent to woher (where from), if there is one?
No, it would have to be var for that. var is for location ("where is it") and vart for direction ("where is it going to").
My understanding is yes; In general, Swedish makes less distinction between the simple present and continuous ("I swim" vs "I am swimming"), just using the present for both. Later on there's a module on Continuous forms for when you need to emphasise that something is going on right now or is continuing to happen. I think Duo always accepts both versions (e.g. "Jag läser snabbt" should accept both "I read quickly" and "I am reading quickly").
It is poor grammar to end a sentence or question with a preposition. These translations are breaking my heart.
Unfortunately, I can't think of a better way of getting across the distinction between var and vart.
Personally, I don't mind ending on a preposition- to my mind, "poor grammar" is something that is not clear and natural, and all the alternatives I can think of are ambiguous ("where is the duck swimming?") or unnatural ("Where to does the duck swim?"). I have no idea if Duo accepts "Whither swims the duck?" but I suppose that would have been the traditional answer.
We try not to enforce prescriptivism. Ending a phrase with a preposition is not inherently ungrammatical, just frowned upon by some.
How can you hear the difference between 'vart' and 'vårt'? Or must we rely on understanding the context because no difference exists?
vart is pronounced roughly as "fart", and vårt is pronounced roughly as "wart", if we stick to the vowels. It's just an approximation but it should at least help showcase the difference.