Kate definitely fancied Spencer. She admitted it to Davina. She was genuinely fond of him just as Jade was genuinely fond of PJ.
Jade was jealous of Kate for the simple reason that all the boys admitted to fancying her. I also think Jade was jealous of Sophie and that's why she resorted to bullying her.
Kate didn't flirt with anyone other than Spencer. She had a laugh with Jonny (a bit like Craig and Claire in BB1) and, after a strained start with Alex, she tried her best to have a laugh with him too.
Sticks, you must stop believing everything you read in the tabloids.