Correct. The original word is (w)orm, cognate to English wyrm (an archaic word for a dragon); the W sound in Scandinavian languages just vanished before back vowels like O a very, very long time ago (pre-Viking, even!).
Danish has been influenced by continental German, where a snake is of course a Schlange and a worm is a Wurm.
Yes, Wurm (or Lindwurm) also means dragon in German, and is part of the germanic folklore. For instance the word "Wurm" can be found in Wagner's Ring Cycle: "Wurmes Gestalt schuf sich der Wilde; in einer Höhle hütet er Alberichs Reif!"
I suspect not, at least not without specifying the snake was venomous. Of course, even a bit from a non-venomous snake can be serious (any puncture wound is potentially bad news, and on top of that snakes tend to have rather fragile teeth, so it's not unusual depending on where and how they bite for fragments of the teeth to break off in the wound, which will then almost certainly end up infected if they aren't removed).