Good but you've confused youki with youkai. Youki in fanfic usually describes the spiritual power that a youkai, or a demon, has while the demon itself, like Sesshoumaru is called a youkai not a youki.
Secondly Kagome is actually a hanyou, like Inuyasha, being a halfbreed having a youkai father and a human mother, or vice versa. She is not an inuyoukai, but she is part inuyoukai. Hanyou can mean any mix of human and youkai lineage regardless of the type of youkai one of the parents is.
Thirdly it would be fairly obvious to Inuyasha and Sesshoumaru that Kagome is a hanyou that being she'd have likely canine ears and slit pupils like Inuyasha. I'm sure that Inuyasha and Sesshy would notice that she's part inuyoukai too. Dogs have an excellent sense of smell. They'd notice that she's more like them than say Kouga whose an ookami or wolf-youkai. Difference in scent.
The Alpha dynamics are interesting, but Sesshy and Inu wouldn't actually be Kagome's Alphas unless she was part of their Pack, which she wouldn't be unless she joined their family. The social hierarchies within teenage social groups don't neccessarily count. They would merely be her seniors if they are older at school, but not have authority over her in the sense of an Alpha, being that neither are her father, who is the Alpha of their family. After their Dad, Souta would be the Beta-Male, being senior to Kagome because he's her older brother, thus he watches over her, even though her Dad is not his Dad.
Atleast that's how I'd envision the Pack Alpha dynamics. You probably have a different idea.
I really like your story so far, and really can't wait to see wear you're going with this. One thing is bugging me though: how is Kagome planning on hiding the fact that she is a half-demon? If everyone knows that Souta, a human, is her half brother, won't it be kind of obvious? I guess we'll just have find out! :) Keep up the good work!