Ok, I'm slightly confused.
I'm guessing when you say that you get local SD and HD channels (NBC, CBS, ABC, PBS, FOX, ABC Family) that you're referring to the entire house located in the USA in general. Not specifically, that those are the only channels available on the the Slingbox, and mention of the "cable channels" is more of a non-sequitur or red herring.
Through clever deduction, I've surmised that your Slingbox Pro-HD is connected to the network via SlingLink. Probably a router is mixed up there, but more importantly you are actually able to connect remotely to the Slingbox from Korea. This would eliminate any sort of network connectivity issues.
However, you never mentioned how your Slingbox is connected to its video source? Is there a RF cable connected to the Slingbox's digital tuner? Or are you using a seperate set-top box? (and if so is it powered on? what manufacturer and model is it? is it connected using component, s-video, composite, ANT IN?) Is your Slingplayer options selected to the appropriate video source?
Since the Slingbox can only rebroadcast what it receives, at this point it sounds like it isn't receiving any signal to rebroadcast. But, maybe if you elaborate more on the questions above then we can try to troubleshoot it together.
Yes, my entire house located in the USA gets the local channels and all of the cable channels that we pay for through Comcast. We have a digital box for all of the TV's in our house from Comcast.
Yes, my slingbox is hooked up through a slinglink. My slingbox is located in the basement and our router is on the main floor.
The slingbox is connected through RF cable. That connects to the Comcast Digital TV Tuner. Yes, it's powered on. Sorry, I'll have to skype with my mom for the exact model #. It is connected through composite cables.
Sorry, that's all I've got for now.
You need to switch the order the cable runs. Your cable box does not work because it can't talk back to the CMTS (cable server). You get the over the air channels in HD because comcast has to let them through. The slingbox does not pass traffic up so the box turns off after a while. For the sling to work you need to take the RF from the wall to the cable box then (if you want) to the sling.
Also if the cable box is off and you have it set up like above. Some will bypass the rf thru. Once you hit the power button you get no signal unless your on channel 3