I think you don't elide "da" before a word with a vowel because then you wouldn't be able to tell if it signified "da" or "di". In this case, the sentence would then be "His father goes in a bee's costume" if you contracted to "d'ape". I could totally be wrong here, but that's what I gather :-/
I agree, "di" can be elided (I couldn't find anything about elision of preposition "da").
"Suo padre va in un costume d'ape. (d' = di)" could also mean "His father goes in a costume made of a bee." :D because preposition "di" can be used to indicate what an object is made of. http://italian.about.com/library/fare/blfare157a.htm (#2)
I've encountered several Italian sentences like that lately. Some where it sounds like two people at talking at once. Others with lots of static. Some, as you mention, where the slow audio appears to say something different. I hope Duo has an audio improvement project! :-)