I saw your other post about the programming - I'm not sure how to tell which broadcast you will get. My guess would be that if the network has one primary game shown it will be that one, regardless of origin and destination. (It could vary for any flight on whether Sunday travelers were coming or going, people have connections, etc...which game would they prefer...not worth Southwest's optimization efforts on something they provide for free.)
So that's my best guess, it will be the default national game. Looks like you have a pretty good shot at it among the alternate games.
Slingbox is most likely a "no" - the terms call out exclusions for VPN or streaming applications.
Considering it takes a good couple minutes to load email, I think it's safe to say Slingbox is a no.
I replied to the other post that the local broadcast channel feed is NYC so you're probably going to get the game that's being shown in NYC, and that's most likely also the national game as @DancingDavidE has suggested.