After getting an Article 15 and having my PCS orders cancelled, how can I get the same orders once my punishment phase ends?

Asking for Korea's shouldn't be hard

All you can do is ask.
Korea is generally not viewed as a desirable assignment, so your odds are decent.

That is entirely up to your branch. When your orders were canceled they more than likely cut orders for some other soldier for that slot. You can attempt to get a hold of branch and ask them after you flags have all been lifted. I warn you though, branch really does not care about the needs or wants of anyone below E7... I've heard of branch telling SSGs to shove it. There are just so many moving parts at that level that finding a way to send a lowly SPC someplace is not high up on the priority list. I got lucky once and had a cool branch manager who got me a PCS to Carson once. I had to work at it and it took me nearly a year of back and forth phone calls and emails, but I got it done. I was also coming off my second deployment and had just been promoted to E5 (both are situations that make it slightly easier to get a move).