"Slow down, please."
To me (native speaker) it doesn't sound like a complete sentence. It would be like saying "please slow", and slow is not a verb in chinese. The “一点儿” here makes the translation closer to "slower please".
What would it sound like to you without the 儿? It would still be grammatically correct, wouldn't it?
Does "please" always have to come first? Why won't the two first characters ('please'+'slow') suffice? What do the last three characters mean? Does the hint for "down" point to a character referring to the direction 'down' (the character was not available for choosing)?
Yes, "please" always has to come first. It would sound incomplete without the last part. The last few words literally translate to "a bit". And I don't understand the last part
as Swansae said, slow is not a verb in chinese, and so the sentence doesn't work gramatically when it's just "please slow". 一点 means "a bit" or "a little" so the sentence roughly translates as "slower, please". 二 is a personal preference and certain dialects use it more than others. Yes, 下 translates to down but can be used differently in different contexts.