I'm about to drill through dry wall to place in screws to hang a coat rack on. I noticed that there is a vent directly up on one of the places I have to drill on (picture shown and drill area marked w/ pencil). Should this be ok, or leave this area alone? Thanks a lot.
Looks like a return cover, which means there's likely no ducting at all (just an open stud bay).
If that's the case, putting a nail/screw/anchor in the cavity, could lead to dust and debris getting snagged up on the object. Will it cause a problem? Probably not, but it surely could.
Since you're looking to hang a coat rack, you should do your best to attach it directly to framing members (studs). Coat racks can be loaded with quite a bit of weight, and are often subject to abuse (yanking, pulling, etc.).