I'm not 100% sure on 買ってこっか. I think it's a shortening of 買って来ようか, but if someone who knows for sure could check I'd appreciate it.
Someone else can correct me on this one but I think you're close. I actually believe she's just using an informal way of saying 買って行こうか. My second guess would be 買ってみようか but it seems more likely it'd be the first. Either way, she's sharing an intent to go buy meatbuns with the viewer/admiral.
