Roxas
Even though Sora has literally next to no idea who Roxas is until Riku mentions it, Kairi does know who he is: voice wise, anyway. The one and only conversation they share (aside from the one with Namine and Sora) basically is there to reconnect Kairi and Sora again.
Kairi is walking with Selphie on Destiny Islands, when she hears a voice: 'Namine?' This attracts her attention and causes her to clutch her eye, as if confused or something along those lines. Roxas then talks to her, and she presses information out of him, since he says he knows her, since she's that 'girl he likes'. Kairi asks for a name, and Roxas gives her his. Kairi then rephrases her question, to ask for the his name; but Roxas never gets around to answering, since Sora answers instead.
Due to them having the one conversation, there isn't much else to add. When Riku mentions that Roxas is Sora's nobody, she shows no recognition at the time. Though, when Roxas appears out of Sora, she treats it pretty well (as if she had met him before, which she had). It also appears Roxas knows her, since he smiles in Sora and Kairi's general direction (it could possibly be Kairi, but that's up to the viewer to decide). The entire 'Kairi/Roxas' relationship is based around the structure of one short conversation. Strange, huh?
Axel
When Kairi's sharing a sentimental moment with herself, Axel appears and disturbs it. At first, he seems rather 'friendly'; until he brings his nobodies along, and sounds rather threatening in the mean time. Which, with the appearance of the Nobody's, would show why Kairi said to him he wasn't 'acting very friendly'.
The reason why Axel turned to Kairi in the first place? Most likely, to find Sora. For Axel, it was probably something along the lines of 'find Kairi, find Sora, find Roxas' since he was still on 'must find Roxas' mode. At first, he seems rather nice to Kairi, but then gets rather annoyed when she doesn't really oblige to coming with him. His speech doesn't really have much affect on her, even mentioning that he would take her to see Sora. When she runs off, he doesn't really say much else.
However, after Kairi finds her way to Twilight Town, Axel turns up. He praises her for her courage, but arouses no response from her. He stops all the protection surrounding Kairi without lifting a finger (since, Hayner and Pence run straight through him and Olette gasps), and grabs hold of her and drags her away. Kairi offers little resistance except for dragging her feet, which won't really help since Axel's most likely a lot stronger than she is.
Despite that fact, according to Axel's later remarks, she does get away from him. How this happens is unknown, since it is never shown or mentioned. When she did escape, she was recaptured by Saix, which probably means she escaped him when Axel was trailing through The World That Never Was. This idea is cited to a point since Axel also mentions Kairi being in the dungeon: he probably saw her get recaptured, or, after talking to Sora the first time and being driven off by Saix, went to find out to gain Sora's trust. Either way, Kairi shows only anger towards him once, and no other emotion, so it is unclear how their relationship is, since she never meets him again.
Selphie
Kairi and Selphie have most likely been pretty good friends since the days in KH1, though they were never actually shown talking (which is probably the same in Tidus/Wakka's case as well). Selphie's the female friend Kairi needed, to be precise, since she hung out with Sora and Riku a lot. It's more than likely since Sora and Riku vanished, Selphie may have been the support Kairi needed while she waited for them.
When Kairi asks Selphie about the boys they used to hang out with, Selphie eagerly replies asking about Riku. But, when Kairi asks about the 'other boy' Selphie has no idea who she is referring to. This is most likely due to Sora regaining his memories, at the time, and Kairi's heart probably only remembers him slightly to the point of a nagging. From the fact Kairi tells Selphie a lot about the letter she wrote, one can determine she would most likely share a lot of things with her. Probably, these two are bordering the 'best friend' mark, even though Kairi's best friends are Riku and Sora.