I see Alex Danvers as more of an ISTJ but I have not yet seen all the episodes, so perhaps I will later see the Intuition argument. From what I have seen, though, i agree with most of your other typings. But again, I need to go back and watch more of the show.
Kara Danvers - ESFJ - Kara is all about protecting people and helping them. She wants to go big but doesn't see the negative effect her actions might have. Despite her planet being destroyed about 2 decades ago she is still very focused on her past and has trouble moving on. Superman is her huge idol and she constantly compares herself to him. I would say her two strongest functions are Fe and Si with Fe being stronger because protecting people is way more important to her than being as good as Superman (when she decides to go small to learn how to be a superhero).
Winn Schott - INTP - As he said it, his whole life is an algorithm. Which is something only a Ti-dom would say. He shows his Ne in his ideas and he's not very experience/opportunities oriented. If it wasn't for Jimmy, he wouldn't decide to tell Kara how he feels.
James Olsen - ESTP - Jimmy is most likely a Se-dom. When he sees an opportunity he takes it and also tries to inspire people around him to do the same (Fe). He often gives advice to his friends and wants them to succeed (which he clearly expresses).
Alex Danvers - ENFJ - She wants to help people, mainly people close to her, just like Kara, however, she uses clear Ni-Se. She can see Kara is not ready to fight crime and trains her to be better. Her intuition is much better than Kara's and she can predict the results of actions. Alex is also a risk taker and willing to sacrifice herself to help people. When she sees an opportunity, she grabs it, like when she joined D.E.O. and later was disappointed when she learned they might have recruited her just for her connection to Kara. Though she seems to be too withdrawn for a Fe-dom, it could be that she doesn't want to cause her more trouble/worries.
Cat Grant - ENTJ - Cat is very emotionally distant and prefers efficiency above everything else. She is very intuitive, one step ahead of others and very logic oriented. Her main way of gathering information is the surrounding world and she prefers to work with facts instead of subjective logic. She is also the only one in the show who managed to figure out Supergirl's identity on her own and directly confronted her about it. She's really good with giving advice and loves perfection.
Alura - IxFP - We know very little about Alura since her screentime is mostly from AI or memories of other characters. Though, I see her as a Fi-dom due to her "betrayal" of her sister to arrest her. It comes of very Fi-Te oriented (this is the most efficient way to do something right). I get Si vibes from her as well (this is my duty and this is how it should be done) but it could come from her Fi rather than her Si.
Astra - ESTJ ? - Astra is a born commander. She is assertive and doesn't let anyone challenge her decisions. She has also very strong sense of what is right and wrong and doesn't have a problem treating Kara as an enemy after she sides with humans. I'm not very sure about her other two functions but she leans to ESTJ because she hoped Kara would become her ally and wanted her to join her army. She also didn't get over her sister betraying her and seems to be very oriented into past. So I'm leaning to ESTJ but I'm not certain.