The broken header appears to be the right speaker connector (or the sleep LED connector).
Comparing it with the "left side" (and considering the two cables only) I tend to the speaker connector. The accompanying text is not very precise.
Both links/pics from: iFixIt's MacBook Unibody Model A1342 Logic Board Replacement
OP, Mac OSX does indeed support BD-R as others have already said, assuming you have a compatible DVD player. Yet, while one of my macs mounts the volumes automatically in Finder, another one won't. Instead, I must run Disk Utility and manually mount the volume. After that, everything works as normal. Hope that helps in the event that is an issue.
I got it fixed by inserting a needle in the case’s holes right underneath the lid. Keep pushing it up while holding the lid with a slight press, repeat the process and keep switching the holes all the way up!