Here are some ideas for Izzie's relationships. I'll edit in the finalized version once I've gotten everyone's input, and post it in the repository after. :)

Alex: I feel like Izzie would think she's amusing. She laughs every time Alex takes a sarcastic comment literally, but points it out to her afterwards. Izzie would take to fondly teasing her, too.

Henry: She's probably heard of the newest farmer in the Valley, but I don't know if they've ever met. Maybe he's been a patron at Daisy's café?

Everett: Izzie frequently goes to the clinic to get her grandfather's medicine, so I feel like she and the doctor should be well acquainted. Because Everett doesn't like exerting himself too much, Izzie would probably volunteer to do anything that requires strenuous physical labor. I can also see her popping in the clinic just to ask him some random medical questions because she's bored.

Eli: Since they're the same age, Izzie would probably want to befriend her. They're opposites to some degree, but I think they have that sort of 'unlikely friends' vibe. Izzie likes to talk, so she'll constantly try to engage her in a conversation. She'll probably try to recruit her in one of practical jokes, too. 

Gaku: I agree with you! I think they'd actually be good friends (and maybe comfortable enough to throw affectionate insults at each other?). They seem to have a lot in common, and I can see them somehow bonding over their affinity for music. Izzie doesn't get fazed by Gaku's tough guy persona (like everything else, she thinks it's amusing), and she can be just as blunt as him, so they don't need necessarily need filters around each other. 

Isis: Izzie likes to visit the flower shop on her free time to look at the flowers and just talk to Isis. The florist is a really good listener and she has an amazingly soothing aura, so she Izzie's go-to when she's feeling unusually down.