While Tatsuki does have the requisite lesbian hairdo her concerns are pretty reasonable in her situation. The Inoue she knows is week and timid and vulnerable. Tatsuki on the other hand is a self-styled protector so it's natural for her to be concerned when her charge goes missing for months. She knows that Ichigo knows more than he's telling about what's happened but he's giving her the cold shoulder. From her perspective this doesn't make any sense because she was his close friend and she was always there for him when he was little so she feels she shouldn't be left out.

I don't think she knows anything about hollows or shinigami beyond the fact that Ichigo hops out of his body from time to time and carries around a big sword. Obviously she would think she could help somehow in the absence of any knowledge or first-hand experience in facing how powerless she would be against a hollow. And even if Ichigo told her something she'd still think she could hack it somehow and would try to fight probably getting herself killed in the process.