Yes! Well spotted. Etymologists think it's probably derived ultimately from Latin vitula, which means "string" if I remember correctly.
Swedish actually used to have the very direct cognate fidla for a fiddle as well. It then turned into the word fela, which you may come across in modern Swedish, if very rarely.
Yes, Swedish always uses the indefinite for playing instruments in general. If you use the definite, it sounds like you're talking about a specific flute or violin.
This doesn't quite extend into a general pattern for all of Swedish, though it does feature in some other constructions as well - e.g. "we can dance the tango" = vi kan dansa tango.
Yes, and it has been accepted for over three years. Your English is not being criticised.
But I'm guessing you had a word tiles exercise and the system decided only to accept the solution it had in mind. If that is the case, we unfortunately cannot control this as course contributors.