wolno is related to wolny(a) in anyway? what would be the "adjective" not conjugated by gender or case for -slow- ?
"Wolno" is an adverb derived from the adjective "wolny".
In another context it may be a defective verb (a verb that has very little conjugation, see here meaning "it is allowed", like: "Tu wolno palić" = "One can smoke here".
As for your question, I think you asked about the adverb... but if not, please precise your question.
It can, but I really can't imagine interpreting this sentence like that...
In this meaning, in order to avoid ambiguity, you should rather use the synonym "swobodnie".
In English, the "do you" in ‘Why do you write slowly?’ can be omitted, making it a functional question, rather than a personal question. Submitted correction.
I am having trouble understanding where to place the adverb. Could it be before the verb here? Or is it different when it is a question? I have in my head something like 'codziennie odwiedzamy naszą rodzinę', where it is adverb+verb, unlike in this exercise
It could be. I don't know if there are any rules. Putting words at the end of the sentence gives more emphasis on them, and in this sentence it seems more logical to accent "slowly".
In the comment section of other sentences it has been said that the adverb should avoid being at the end (eg dobrze wychowujemy nasze dzieci as opposed to wychowujemy nasze dzieci dobrze). How come it is not the case for this question and wolno is allowed to be at the end? Is it because it is a question? Would a statement have to be 'wolno piszecie' rather than 'piszecie wolno'? Thank you.