The association between gender and the noun is in large part (albeit not completely) arbitrary. In this case, since Halter is a "masculine" noun, the compound Büstenhalter is "masculine" too. So it gets the "masculine" article der.
If it helps, instead of looking at German genders as "masculine vs. feminine vs. neuter", look at them as "der gender vs. die gender vs. das gender".
Of course they do. There is no actual rule to it. You just have to know. Often words ending in "er" are male, but not all of them. It's one of the reasons German is so difficult. Just avoid it. English is easy and efficient.
I kinda wish this would be true because that word looks and sounds funny. But I'm german and I know it's not the actual word. If it would have said that this is the dutch or danish word I would bei more convinced to believe that this could be the actual word.