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.).
Open the vent and see which way the duct goes from the vent. It'll turn up, down, left, or right. That'll tell you where it is and if you can drill under it.