Your daughter sounds advanced, and I think you should encourage that. I would get her materials at home which can help her learn. There are a lot of different sorts of things you can do for different subject areas (e.g. Singapore Math).
She must be getting concepts like odd and even numbers from somewhere, perhaps from child educational TV shows. Providing her with more will just let her advance more quickly. I would probably start at around a second-to-third-grade level with some tool that lets you assess her weak and strong areas, if the goal is to make sure she's a good fit for a future acceleration or just to make sure she's well-rounded. I use the IXL online website for this, since it's tied to state standards.
No matter what, I think you have to ensure she's provided with something. No one can learn in a vacuum. No matter how good she is at picking things up by osmosis, she won't get a well-rounded foundation in math or anything else that way.
My son's kindergarten teacher wound up recommending against a grade skip for reasons of maturity as well (another young kindergartner here). My son currently gets third grade math sent home (he is in a half-day kindergarten) and advanced reading in school, which is not perfect but an okay start for this year. The teacher is saving his work for an end-of-year review which may include advancement.
With the multiplication, I would probably just print her out a times table (try to find one with number relationships highlighted, like square numbers) and drop it in her path somewhere. I would still make sure she has fuller exposure to level-appropriate math materials, though.
She sounds like she has a good imagination. I wouldn't worry about it. Kids will often pick on anyone who seems different. Your daughter will seem different at times. It's just part of growing up, and I don't see it as a problem, nor anything that needs to be changed about her.
About the panic attacks, I guess if you think it's a problem, I'd talk to someone about it who's an expert in the field. I tend to think it just sounds like a sensitive kid, but I am not a doctor and don't know the full situation either.
