It doesn't matter what you look like. You can look white and yet have a non-white ethnicity. The box isn't asking what you look like - it's asking what your ethnicity is. Deciding someone's ethnicity based on their appearance is racist. Genes play out in interesting ways, especially where there has been mixing in the more recent gene pool - you could be majority one ethnicity but due to which genes are more dominant, look more like one ethnicity of which you don't have much background.
Plus, there have been cases documented where one person's family are understood to belong to the same ethnicity going quite far back, but they don't look like any of their relatives. Having an unusual appearance for one ethnicity doesn't mean you don't belong to that ethnicity.
When you're mixed race, there is often a disconnect between your actual ethnicity and how people perceive you, because there are lots of racists out there. Take Meghan Markle. I don't especially like the woman, but I do sympathise with her on account of the racism that she has encountered because people have taken one look at her, and decided her ethnicity for her. That's not how genetics work...
I do also think if people treat you one way based on your appearance, it may shape what you identify as. But that, again, doesn't change your ethnicity. Your identity is not the same thing. Your identity is - but not always - often more linked to which culture resonates more strongly with you, and/or where you grew up.
Mixed race is a perfectly neutral term - I use it to describe myself because I am mixed race. I don't know why on earth you thought you had to introduce some extra offensive words to the thread.
If one parent is Black African and one parent is White, you are clearly not Black African. You are mixed race. When you are 50/50 one thing, you cannot be majority one thing. It's simple maths.
How far back do you go? Well, I'd go as far back as you know. For some people, that might just be as far back as their parents or grandparents. I don't think you need to research your entire family tree as TBH, if you were to go 100s of years back, pretty much everyone would have to say they're mixed race. It's a matter of sensible rounding.
If both your parents would sensibly tick the same box, you fully belong to that box. If one of your parents would tick another box, you cannot claim to be fully one thing, so you go to mixed race. Or, and I have made this point previously, you could tick 'prefer not to say' or 'other.' What you don't do is pick a box that you like the sound of.
I'm not sure why you brought Jewish people into this argument either. By modern definitions, being Jewish is not an ethnicity anymore - it's definitely an identity, but not necessarily a racial identity. I mean, you can convert to become Jewish, and you cannot convert from one ethnicity to another. Thus, Jewish people have the whole range of boxes to choose from.