"Please call a doctor now!"
I missed the はやく. The sentence used 'now' and I thought it was いますぐ.
Given the flexibility of word order, is it more natural to put the adverb before the verb, or at the start of the sentence? Does it change the emphasis?
I don't think kudasai is necessary. If you are telling people to "call a doctor now!" I doubt you would add please.
For the English sentence yes, but not quite for Japanese. The use of polite form is largely depending on the relationship of the speakers rather than the emergency. I would expect a Japanese to say Kudasai when saying this to, for an example, a nurse. Also dropping Please leaves an imperatve order in English but dropping Kudasai does not yield an imperatve order in Japanese but a soft familiar request. So they are not a direct comparison.