I want to pass my ethernet, speaker and sub wires into my newly built media/bar cabinet, but I hate having to pass them under the door, so I'd like to drill a hole as shown by the green circle in the image.
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Not a bad idea what so ever. However, let me clarify on the posted answer... A hole saw can be used to cut clean perfect holes, but are limited in cut depth. You'll most likely need to use the hole saw on both sides of the jamb.
A hole hog, as another commenter mentioned, is also known as an auger bit. It has a pointed threaded tip that will pull the bit thru whatever material you're trying to drill thru until the tip can't pull anymore. From there you'll need a little extra force to get the bit to finish it's cut.
Both are feasible tools for what you're looking to do.
I would suggest getting a hole cutter instead of drilling a hole so the hole can be future proofed if you want to pass more wires someday.
Probably too late but I would actually go through the wall between the molding and the corner. Measure things on the side from the picture, transpose the hole location into the other room and drill from the other room. 3/4" auger or installer bit should fit fine. That's just me.
Drilling the frame will end up being like 5" of solid wood counting the doubled stud and the molding, drilling into that space will be just wall material/hollow cavity/wall material. It will also be less likely to cause issues in the future. Need larger hole? Elongate with a jab saw, and use those plastic cable hiders to cover the hole.